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00 x { x-00 2 if1· f x > 00 ' =
r-+
X
=
then 1(f) = -00 and 1(g) = 00, whence 1(f) + 1(g) is not defined. ) c ) If 1(f) 00, the result is automatic. If 1(f) < 00, then =
1 (f) + 1 ( - I )
is defined, whence
0 = 1 (0) = 1 [1 + (-f)]
:s;
1 (f) + 1 ( - I ) = 1(f) - [- 1( - I )]
=
1 (f) - l (f);
d ) Since f :s; f, if f E Lu, then 1 (f) :s; J(f) . If f :s; g E Lu, then
J(f)
:s;
J(g),
whence J(f) :s; 1(f) : 1(f ) = J(f) . Owing to the linearity of I and the last equality, if g E L, 1( -g) = J( -g)
=
I( -g) = -I(g).
Thus l (g) = -1(-g) = - [-I(g)] = I(g) . If f E Lu and L 3 gn t f, then 1 (gn ) = I (gn ) :s; J( f) and so
l (1 )
:s;
1(f) = J(f)
=
I nlim --+ (gn ) ex)
=
lim-+ CXJ 1 (gn ) n-
:s;
l (f) :
1(f) = l (f) = J(f) . e) If any 1 (fn ) = 00, the implication is automatic. If each 1 (fn ) is finite and E 0, then for some gn in Lu,
>
Moreover, by virtue of 2.1.4d ) , g
J(g) =
00
2:
n= 1
00
�f 2: gn E Lu and
J (gn )
n= 1 :s;
00
2: 1 (fn ) + E.
n= 1
50
Chapter 2. Integration
Furthermore, g 2':
00
I and hence 1(1) :::; I(g) :::; nL= 1 1 (In)
+ E.
D
[ 2.1. 7 Note. Although the implication
{ I 2': � In}
*
{
1(1 ) 2': 1
(t,ln) }
is valid, the implication
is not.
] -x
I IN IS IN L 1 IFF -00 < (I ) 1 (1) < 00. FOR IN L 1 , J (I ) IS THE COMMON VALUE OF (I) AND 1 (I ) . THE I MAP J : L 1 I J ( I) IS THE DANIELL-LEBESGUE-STONE (DLS ) FUNC 2.1.8 DEFINITION. A FUNCTION
ffi.
=
I
3
1
r-+
TIONAL. ( If I E Lu , then 1( 1 ) > -00. Hence, if I E Lu and 1( 1 ) < 00, then -00 < 1( 1) l( l ) = 1 ( 1 ) 1( 1 ) < 00, i.e., if E Lu and 1(1) < 00, then I E L1 . ) The properties listed in 2.1.6 for 1 lead to =
I
=
2.1.9 THEOREM. a) L C L 1 AND I; b) L 1 IS A VECTOR SPACE AND IS LINEAR ON L 1 ; c ) I E L 1 IFF FOR EVERY POSITIVE E AND SOME g AND -h IN Lu , h :::; g AND I[g + (-h) < E; d ) L 1 IS A FUNCTION LATTICE; e ) THE MAP IS A NONNEGATIVE (LINEAR) FUNCTIONAL ON L 1 ; f ) IF L 1 3 t AND FOR ALL n , :::; M < 00 , THEN E U AND t CONVERSELY , IF I E L 1 AND L 1 3 t THEN
JI L =
J
J (ln) J(I) . In I
]
I J:::;
J (In) In I , l J (In) J (I) . PROOF. a If I E L and gn clef I, E then L gn I , whence I E Lu: L Ln · Furthermore, 1(1 ) 1( 1 ) : Il L = I, whence 00 1 ( 1 ) = 1( 1 ) l(l) -00 : I E L 1 . A similar argument shows that JI L = I. t
C
)
= =
>
n
=
3
N,
>
t
Section 2.1. Daniell-Lebesgue-Stone Integration
51
L l , C E JR., then { l1((cfcl)) == inf-1Lu(-3h?:cl) f=J(ch)[-c1(1)] = c infLu3h?:f J(h) = c1 (1) if c 2': 0 = c1 (1) = cl ( l ) if c < 0 ' whence cf E L I . If {f, g } e L I , then the preceding argument and the subadditivity of 1 imply 1 ( 1 + g) � 1 (1) + 1 (g) = J(I) + J(g), -l(f + g) = 1 (- f g) � - 1 (1) - 1(g) = -J(I) - J(g ) , 1 (1 + g) 2': l ( l + g) 2': J(I) + J (g) 2': 1 (1 + g), whence J is a linear functional on L l . c) If f E L l and E > 0, then for some g in Lu, g 2': f and E I(g) < J(I) + 2 . E Similarly for some -h in Lu, -f � -h and I(-h) < J(-f) + 2 . Hence h :::; f :::; g and J[g + (-h)] = J(g) + J(-h ) :::; J (I) + J(-I) + E = J(O) + E E. Conversely, if there are g and - h as described, then, as noted after the introduction of Lu, for any k in Lu, -00 < J(k) . Since I(g + (-h) = I(g) + I(-h) < E, both I(g) < 00 and I( -h) < 00. Hence 1 (1 ) � l(g) < 00, 0 :::; 1 (1) - l ( l ) = 1 (1) - [ - 1 (- I)] , � J(g) + i(-h) = J[g + (-h)] < E. d) If Lu 3 h i :::; Ii :::; gi E Lu, i = 1, 2, then b) If f E
-
-
=
-
-
-
-
-
e L I and gi and hi , i = 1, 2, are chosen as in c) , then - ( h l � h2 ) -h I � - h2 E Lu, J (g
K c g� u, 1) =
n On ·
nEN
(
n+ 1 -- l, l n
) cle=f On E O (X); b) On E K(X);
c)
In sum, every compact set K is contained in a
compact Cij . [ 2.3.4 Remark. When X is a locally compact space, KGij is the set of all compact sets each of which is a Cij . Some define Sb(X) to be the a-ring generated by KGij: Sb(X) �f aR (KGij) and some define S,(3(X) to be aR[K(X)]. The reader is encouraged to explore the relations among these definitions and those used in this book.]
D
2.3.5 Example. In 2.1.22, if X �f JR., consists of all functions that supp U) is empty, finite, or countable while
{J E Ll } {}
ttI:
{
UE
D} /\
{�I I(X) I } } .
I
such
< 00
Consequently, JR. D. The function JR. '3 x r-+ x E JR. is such that # [supp U)] > No, whence although for each a in JR., E= U, a) = {a} E D (cf. 2.2.22). 2.3.6 Example. In its customary topology, again JR. is a locally compact �f I(x) dx (the Riemann Hausdorff space. If I E Coo(JR., JR.) and integral of Urysohn's Lemma implies that aR[K(JR.)] = D.
I tt D,
I),
1U)
l
Chapter 2. Integration
78
2.3.7 Exercise. In 2.3.2: a) if E E 5,6,
p,(E) = inf { p,(U) : E c U, U E O (X) } ; b) if p,(E) < 00, then p,(E) = sup { p,(K) : K c E, K E K(X) } . The conclusions a) and b) above motivate the following terminology. 2.3.8 DEFINITION. FOR A MEASURE SPACE (X, 5, p,) , A SET E IN 5 IS outer regular (inner regular) IFF
p,(E) = inf { p,(U) : E C U E O (X) } , (p,(E) = sup { p,(K) : E ::J K E K(X) } ) . AN E THAT I S BOTH OUTER REGULAR AND INNER REGULAR I S regular. WHEN EVERY E IN 5 IS OUTER REGULAR (INNER REGULAR) (REGULAR) , (X, 5, p,) AND P, ARE OUTER REGULAR (INNER REGULAR) (REGULAR) . 2.3.9 Exercise. In 2.3.2, if X �f [0 , 1] in its customary topology and I is a) p, = A (cf. 2.2.40 ) ; b) if E E 5,6([0 , 1] ) and the Riemann integral r ira.l] x E JR., then :
X
+E
clef { x + Y =
: Y E E } E 5,6([0 , 1]), A (E) = A ( X + E )
(5,6([0 , 1]) and A are translation-invariant) ; c ) if {E, F } c 5,6, ° < A (E) . A(F) < 00, and x E JR., then f (x ) � A[E n (x + F ) ] = d) E
-
l
X F (Y - x) X E (Y) dy and f is continuous;
F �f { x - y : x E E, y E F } , contains a neighborhood of zero.
[Hint: If E is an interval, b) and c) are valid. For d), c) applies.]
[ 2.3.10 Note. In the context of 2.3.9, the measure space is [JR., 5), (JR.)), A] and A is Lebesgue measure. The sets in 5), [JR.)] are
the Lebesgue measurable sets. Corresponding definitions apply for the notions of Lebesgue measurable functions, Lebesgue integrable functions, Lebesgue integrals, etc.] 2.3.11 Exercise. (Vitali-Caratheodory) In the context and notation of 2.3.2, the conclusion in 2.2.56 obtains. [Hint: Urysohn's Lemma ( 1. 2.41 ) applies.]
Section
2.4 .
79
Complex-valued Functions
2.4. Complex-valued Functions
Little difficulty and much advantage follow from admitting C-valued func tions to the discussion. From this point forward, functions with numerical ranges are to be assumed as C-valued unless the contrary is stated. The image "(* �f "((lR.) of the curve "( : lR.
is { z
Izl
=
'3 t r-+
1 - t 2 . 2t C 1 + t2 + l 1 + t2 E
( 2.4. 1 )
1, z i- - I } �f '][' \ { - I }. The function fJ : lR. '3 t r-+ fJ(t) cle=f
2 it -2 dx o
1+
X
( 2. 4.2)
is continuous and strictly monotonely increasing. For some (finite) number, denoted 7r , ---+lim fJ( t) = ± 7r: "( is rectifiable and t ± oo length of "(
�f £("()
=
27r .
The inverse of fJ is the function t : ( -7r , 7r ) '3 fJ r-+ t(fJ) �E ( -00, (0 ) :
t(O)
=
t(fJ) = 00, lim t(fJ) = -00, t(±7r) �f ±oo . 0, lim Ot7r O.j.-7r
(2.4.3)
The two trigonometric functions,
1 - t(fJ) 2 1 + t(fJ) . : [ -7r , 7r] '3 fJ r-+ 2t(fJ) sm 1 + t(fJ) 2 ' cos : [ -7r , 7r] '3 fJ r-+ -----'--'-:2 :-
(2.4.4) (2.4.5 )
are infinitely differentiable on ( -7r , 7r ) and
cos' fJ = - sin fJ, sin' fJ = cos fJ.
The formal
fJ 2 fJ4 1- -+ - -···' 2! 4! fJ-3 fJ5 fJ - + - · · · 3! 5!
( 2. 4 .6 ) (2.4.7)
converge for all fJ in C. The remainder formulCE associated with the Maclau rin polynomials for cos fJ and sin fJ show that ( 2.4.6 ) resp. (2.4.7) represent
Chapter 2. Integration
80
cos fJ resp. sin fJ on (-7r, 7r) and define cos fJ and sin fJ throughout C. Com bined, they yield
(i fJ ) n cos fJ + i sin fJ = 1 + � � n= l n!
and the definition
�f exp(ifJ) (Euler's formula)
00 fJn exp(fJ) cle=f 1 + L ,.
(2.4.8)
n= l n.
The right member of (2.4.8) converges throughout C. On JR., the series (2.4.6)-(2.4.8) represent three infinitely differentiable functions mapping JR. into JR.. (By virtue of the argument in 5.3.2, they are infinitely differentiable throughout C. ) Direct calculations using (2.4.8) show exp(u + v ) = exp(u) exp(v) ,
(2.4.9)
whence if e � exp(I), successively, exp(n) = en , n E N, exp(m) = e m , m E Z, exp r = eT, r E Q.
(2.4. 10)
Owing to the continuity of exp, the definition e O �f exp( fJ) , fJ E C, is con sistent with the formulre in (2.4.10) . If fJ E JR., I e iO I = ( cos 2 fJ + sin2 fJ) "2 = 1. By virtue of (2.4.3) , 1
whence, for
k
k
e 7ri = cos 7r + i sin 7r = - 1,
in Z, e 2 7ri = 1. If fJ E C, then k
{ COS(fJ . (fJ sm
+
+
(2.4.11)
2k7r)
k 2 7r )
} { cossm fJfJ } . =
.
If ¢ E JR., for a unique in Z, fJ � ¢ + 2k7r E (-7r, 7rJ . If cos ¢ = 1, then cos fJ = 1, i.e., t( fJ ) = fJ = 0, ¢ E 2Z7r . If Z � x + iy and e Z = 1, then x = 0 and e iy = 1 whence e Z = 1 iff Z E 2Z7r i. (The last conclusion is alternatively deducible from the formula fJ 2 fJ2 + fJ6 fJ 2 + . . . cos fJ = 1 1 1 _
(2 (
_
12
)
6!
(
_
56
)
)
applied when fJ E [- 7r, 7rJ .) 2.4.12 Exercise. The least positive period of both cos and sin is 27r. [Hint: sin fJ = - cos' fJ.J 2.4.13 Exercise. a) exp l lR is a strictly monotonely increasing function; b) the function inverse to the exponential function exp on JR. is the logarithmic function In : (0, (0 ) '3 Y r-+ In (y) E JR., i.e., exp 0 In (0, 00) '3 Y r-+ y, In 0 exp : JR. '3 x r-+ x ; :
81
Section 2.4. Complex-valued Functions
lim exp ( x ) c) lim exp(x ) = 00, X4-00 X400
eZ
=
0; d) if z = x + iy,
(cos y + i sin y ) ;
= eX
e) for z in C \ {a} and some unique fJ in ( - 7r , 7r] , Z = exp(ln I z l + ifJ); e) exp' = exp o
�f
[ 2.4. 14 Note. For ,,( as in (2.4.1), if z x + iy E "( * , there is in ( - 7r , 7r ) a unique 8(z) such that e i8 ( z ) = z. Hence, if e = z, then e - i8 ( z ) = 1 and so for some k in Z, ¢ = i [8(z) + 2 k 7r] . The signum function sgn is defined by: sgn : C '3 z r-+
Izl { cl sgn (z) f Z �
O
if z :;to O otherwise
Hence: a) z · sgn (z) = I z l ; b) sgn is continuous on C \ {O}; c)
{
if z :;to O I sgn ( z ) 1 = I otherwise. o
The unit circle { z : I z l = I } ( "( * u { - I } ) is the same as 'lI'. If z E (C \ {O} and z = I z l e i iJ , - 7r < fJ � 7r, the half-line =
[0
clef { =
w
:
w =
re'· 0 , r 2: 0 }
meets 'lI' in exactly one point, which is sgn (z) . If U is open in C, u n 'lI' is relatively open in 'lI' and sgn
-1
- 1 (U) = { SS Ucl=ef{O}{ z
[0 n U n 'lI' :;to (/) }
if 0 tic U otherwise
�f
Hence sgn (U) is either open or the union of an open set and a sin gle point: sgn is (aR[O(C)] , aR [O(C))]-measurable. When z x + iy and sgn (z) u(x, y) + iv(x, y) , then u, v E D. For X, L, and I in the DLS development, a function
�f
f : X '3 x r-+ f ( x ) = SRf ( x ) + i':Sf(x)
�f u(x) + iv (x) E C
is defined to be DLS measurable, D-measurable, or Caratheodory measur able iff u and v are: f E L I iff both {u, v } e L l , in which event,
Ilf ll l
�f J( l f l ) .
82
Chapter 2. Integration
For a measure space
(X, S, I-l), by definition,
{ I �f u + iv E S } {} {{ u, v } C S} ,
{J E L 1 (X, I-l) } {} { III E L l (X, I-l) } . In the circumstances , smce ma:x{lul, I v l } � ';u2 + v 2 � lui + l v i , I E resp. I E iff
L 1 (X, I-l)
J(lul) + J(lv l ) < 00 resp.
Ll
lx , u , dl-l + lx , v , dl-l < 00 .
Unless the contrary is stated, henceforth, functions will be assumed to be C-valued. 2.4.15 Exercise. The map JR. '3 x r-+ exp(27rix) E 'lI' is a continuous open epimorphism of the additive group JR. onto the multiplicative group 'lI'. (In particular, if k E Z, then exp((2k + 1)7ri) = - 1.)
}
2.4.16 Exercise. If z i- 0: a) Arg (z) �f { (J : (J E JR., z = I z l e i O i- 0 ; b) . . (J l - (J2 E Z. #[Arg (z)] = No ; c) { (J l , (J2 } C Arg (z) Imphes 27r 2.4.17 Exercise. If -7r < (J � 7r: a) cP : C \ [0 '3 z r-+ B[sgn (z)] is a non Arg (w). constant continuous map; b) cP (C \ (0 ) c
U
w EC\(e
2.4.18 THEOREM. a) IF THE CURVE "( : [0, 1] '3 t r-+ "((t) E C IS NONCON STANT AND 0 tt "(* , FOR SOME NONCONSTANT cP, cP o "( :
[0, 1) '3 t r-+
U
Arg (w - o )
wey'
IS CONTINUOUS. b) A MAP '1jJ DEFINED ON ,,([[0, 1)] AND SATISFYING
'1jJ 0 "( : [0, 1) '3 t r-+
U
wE"!'
Arg (w - 0 )
(2.4. 19)
IS CONTINUOUS IFF FOR SOME m IN Z, '1jJ 0 "( - cP 0 "( = 2nm.
PROOF. a) For the curve ;y �f "( - 0, if ° = to < t l < . . . < tn
=
1 and
is sufficiently small (and positive), the arc ;Y [tk- l , t k ] is contained in some C \ [0 where 2.4.17 applies: there is a cPk such that cPk o ;Y is continuous on [tk - l , tk ] . Owing to 2.4. 16c) , for some m l in Z, cPl o ;Y (t t ) = cP2 o ;y (t t ) + 27rm l , for some m2 in Z, c/J2 o ;Y ( t 2 ) = cP3 o ;y ( t2 ) + 27rm2 ,
Section 2.4. Complex-valued Functions
83
etc. Thus if if t E [0, t I ) if t E [t l , t2 ) if t E [tn - I , 1) then 1> is continuous on [0, 1) and if t E [tk - l , tk ), then
Since the correspondence ;Y(t)
+-+
l'(t) is bijective, the equation
defines the required ¢. b) If m E Z, then 27rm + ¢ 0 l' is continuous on [0, 1) and satisfies (2.4.19). Conversely, since the map '1jJ 0 l' ¢ 0 l' is continuous, the result in 2.4.16c) implies that '1jJ 0 l' ¢ 0 l' is 27rZ-valued. Hence '1jJ 0 l' ¢ 0 l' is a constant. [] 2.4.20 Exercise. In the context of 2.4.18: a) If r5 is sufficiently small (and positive), then ¢ 0 1' is monotone on each interval [t k- l , tk) ' b) For A ¢ 0 1'(0) . respect to a, . d ( a ) cle cle . A =f hm ¢ 0 l' ( t ) =f , the . dex of l' wzth 'Y �l 27r is in Z. c) On each component of C \ 1' * , ind 'Y ( a) is a continuous (hence constant) function of a. d) Only one component of C \ 1' * is unbounded. e) If a lies in the unbounded component of C \ 1'* , then ind 'Y(a) = 0. [Hint: e) If E > ° and 10'1 is large enough, then in the notations of 2.4.18, sup 1> o ;y < E . ] t E [a,l ) When z E C \ {O}, the count ably infinite set In(l z l ) + Arg (z) is de noted Ln (z). 2.4.21 Exercise. a) If 1' : [0, 1] 3 t r-+ l'(t) E C is continuous and ° tt 1'* , for some ,£ defined on 1'[[0, 1)], ,£ 0 l' : (0, 1) 3 t r-+ L n (w) is continE 'Y uous. b) If z i- ° and eW = z, then w E Ln (z) . c) If t in [0, 1), then e£o 'Y ( t ) = l'(t). [Hint: The argument in the proof of 2.4.18 applies.] There are profound connections between the map h, a} r-+ ind 'Y (a) and basic topology, e.g., the Jordan Curve Theorem , Brouwer degree of a map, etc. A dense but useful reference here is [Sp] where an extensive -
-
, Ill
-
-
m
l I
wU
'
Chapter 2. Integration
84
bibliography is offered. The discussion provided above for the basic topics of this subject is adequate for the current and later purposes of this book. 2. 5 . Miscellaneous Exercises
A A
{
¥A} 2.5.1 Exercise. a) If the set of indivisible elements of a-ring 5 is finite and U is their union, �f { A \ U : A E 5 } is a a-ring and no element of is indivisible. b) If is infinite, #(5) No. c) If 5 is infinite, then is infinite and #(5) No. In sum, if 5 is a a-ring, then #(5) i- No. [Hint: If A i- some nonempty S-set B is a proper subset of A.] An element in a a-ring 5 is defined to be indivisible iff every proper 5-subset B of is 0, i.e., 5 '3 B ::::} {B = 0 } . 5
5
>
5 '3
I
I
5
>
0,
2.5.2 Exercise. For a group G, a function lattice L contained in JR.G , a DLS functional 1 : L r-+ JR., an I in JR.G , and an a in G, if the left a translate of I is I[a] : G '3 x r-+ I(ax), and for each a in G and each I in L, 1 (f[a] ) = 1( f ) , then: a) for all I in D, I[a] E D; b)
1 { f E L 1 } ::::} { { f[a] E L } !\ {J (f[a] )
=
J (f) } } ;
c) {E E D} ::::} {{aE E D} !\ {t-t (aE) = I-l(E) }} (cf. 2.3.9) . 2.5.3 Exercise. If I E L 1 ( X, I-l) and n---+ lim= I-l (En ) = 0, then
r nlim --+CXJ }En
ill dl-l = o.
[Hint: If I is a simple function the result is a consequence of the nonnegativity of I-l. For the general case the density of the set of simple functions in L 1 (X, I-l) applies.] 2.5.4 Exercise. If n : A '3 ), r-+ n(),) E !fj(X) is a net, then lim X ' ) = x-n and lim Xn ( A' ) .AEA .A EA n ( A
=
X . !l
2.5.5 Exercise. An I in JR.[O.1 ] Riemann integrable iff: a) for some finite M and all x , II (x) I � M and b) the Lebesgue measure of the set Discont ( f ) of discontinuities of I is zero: ),[Discont (f)] = O. 2.5.6 Exercise. If 5 is a a-algebra contained in !fj(X ),
h : JR.2 '3 (x, y) r-+ h(x, y) E JR.
Section 2.5. Miscellaneous Exercises
85
is continuous, and Ii : X '3 x r-+ Ji (x) E JR., i = 1, 2 , are S-measurable, then H �f h (II , h ) is S-measurable. [Hint: a) The set E< (h, a) is an open subset U of JR.2 ; b) E< (H, a) consists of all x such that [II (x) , h (x)] E U; c) U is a union of (count ably many) pairwise disjoint half-open rectangles
2.5.7 Exercise. If { 1, J} C D, then sgn (I) E D. [Hint: The set E �f E= (I, 0) is in 5 and I (X \ E) c C \ {O}; C \ {O} '3 w r-+ sgn (w ) is continuous, whence sgn (I) is measur able on X \ E.] 2.5.8 Exercise. If 5 is a a-ring contained in !fj(X), S is the associ ated set of S-measurable functions, and I E S, there is in S a (J such that
I(x) == II(x) l e iO ( x) .
2.5.9 Exercise. a) If 5 is a a-ring contained in !fj(X), S is the associated set of S-measurable functions, and I E S, then I II E S. b) The converse of a) is false. c) {J E L 1 } {} {{J E S} !\ { I I I E L 1 } } .
[Hint: For b) , if E E !fj(X) \ S ,(3 (JR.) , then
I
X E - X (lR\E)
I
==
1.]
Ix
2.5.10 Exercise. What is the result of applying the DLS procedure to the lattice L �f L 1 (X, p,) and the functional I L '3 I r-+ I dp,? :
2.5.11 Exercise. If R is a ring of sets and is monotone, i.e.,
{ { {En }nEN C R } !\ {En C En + d }
'*
{ { {En }nEN C R} !\ {En ::J En + d }
'*
{U {n
nEN
nEN
} }
En E R , EN E R ,
then R is a a-ring. 2.5.12 Exercise. For a a-ring 5, if {In } nEN is contained in the corre sponding set S of S-measurable functions, each of lim In , lim In , and n n (when it exists) lim In is in S.
n---+ =
--+ CXJ
--+ CXJ
2.5.13 Exercise. If: a) (X, S, p,) is a measure space; b) X is totally finite, i.e., X E S and p,(X) is finite; c) p,* is the induced outer measure; d)
Chapter 2. Integration
86
(X, C, ll) is the measure space for the a-algebra of Caratheodory measurable sets, then C = 5 (the completion of S). [Hint: If E c F E 5 and p,(F) = 0, then p,* (E) = 0: 5 c C. If A E C, then for sequences {An } nEN and {Bn } n EN contained in S, E C An ::) An +l , n E N, E ::) Bn C Bn +l , n E N, and
fI ( A) = p,* (A) = nlim --+ p, (An ) = nlim --+ CXJ p, (Bn ) .] ex)
2.5.14 Exercise. In 2.5.13 the conclusion remains valid if X is the countable union of sets of finite measure, i.e., if X is totally a-finite. 2.5.15 Exercise. For a curve "( : [0, 1] '3 t r-+ "((t) E C and an a not in "(* : a) for some positive J, o inf h(t) - 0' 1 2: J and :s;t:S; 1
b) if 0 = t l < t2 < . . . < tn = 1, sup tk - tk- l < J, and :S; k :S;n
2
then for some m in Z,
e
n
� 2)'h = 2m7r; c) ind ')'(O') = m. k= 2
2.5.16 Exercise. If f E JRlR , f OR.) C [-00, (0 ) , and f is usc: a) f is Lebesgue measurable; b) A(E) < 00 implies either f ( x) dx E lR. or, by
Ie
Ie
abuse of notation, f(x) dx = -00. Corresponding statements are valid when f(lR.) C (-00, 00] . 2.5.17 Exercise. If X is a set and {En} nEN C !fJ(X), then: a) There are clef clef 1 1· m - En = E resp. 1·1m En = E such th at sets -
n---+ oo
n---+ =
lim
n--+CXJ b) lim En = { x
X E = XE n
_
and lim
n--+CXJ
XE = XE . n -
=
n U
X
is in infinitely many En } =
X
is in all but finitely many En } =
mEN n=m
En , =
U n En·
mEN n= m
87
Section 2.5. Miscellaneous Exercises
c) lim En C lim En . d) If 5 is a a-ring and { En } n E N C 5, then n n --+ CXJ
--+ CXJ
lim En E 5 and lim En E 5 . n---+ =
2.5.18 Exercise. If (X, 5, p,) is a measure space and £ is the set of simple s dp, E JR., how does the completion of £ with refunctions s such that
Ix
spect to the metric J : £ 2 '3 {J, g} r-+ J ( f, g)
�f
L l (X, p,)?
Ix I f
-
g l dp, compare with
2.5.19 Exercise. (Egorov) If: a) (X, 5 , p, ) is totally finite; b)
x, fn (x) f(x); x ])
c) for each -+ and d) E > 0 , there is in 5 a set E such that p,(X \ E) < E and fn l E � f i E (v. [GeO]). For each in JR., n--lim = 0 yet if 00 > A ( E ) > 0, on JR. \ E, -+ = X ( rn , n +l then X ( rn , n +l fi o.
[Hint: If ENm �f
P,
(ENm ) < ET m .]
] ) (x)
9 { x : Ifn (x) - f(x)1 2: � }, for large
n N
N,
The symmetric difference Al1B of two sets is (A \ B) U (B \ A) . 2.5.20 Exercise. If (X, 5, p,) is a measure space: a) 5 is closed with respect to the formation of symmetric differences; b) for elements A and B of 5, the relation {A rv B} {} {p,(Al1B) = O} is an equivalence relation. The rv-equivalence class containing A is denoted A� .
�f 51 rv of equivalence
2.5.21 Exercise. For rv as in 2.5.20, the set 5� classes, the map
is well-defined, i.e., independent of the choice of the representatives A resp. B of A� resp. B� . Furthermore, p is a metric in 5�. 2.5.22 Exercise. In the context of 2.5.20 and 2.5.21, if
X �f JR., 5 �f 5 ,(3 ( JR.) ,
P,
�f
A
,
then (5-; p) is not a complete metric space. Its p-completion is D, i.e., 5-is p-dense in 5.
88
Chapter 2. Integration
2.5.23 Exercise. If X is a set, S C !fj(X) , and M
then aR(S) =
U
�f { S a
: S a C S, # (S a ) � No } ,
aR (S a ) .
[Hint: The right member of the preceding equation is a a-ring.] 2.5.24 Exercise. If E E S),(ffi.) , then I : ffi. '3 t r-+ I(t) � >. ([0, t) n E) is continuous. The previous assertion is valid if [0, t) is replaced, for any a in [-00, (0) by any of [a, t) , (a, t), (a, t] , [a, t] , or by any of the last four when a and t are interchanged. If g : ffi. '3 t r-+ g( t) E ffi. is Lebesgue measurable and t in the first sentence is replaced by g(t) is I : t r-+ >. { [a, g(t) ] n E} Lebesgue measurable? 2.5.25 Exercise. a) For measure spaces (X, S, f.-ln ) , n E N, such that f.-ln � f.-ln +l , f.-l � sup f.-ln is a measure. b) For (ffi., S)" >.) if f.-ln �f .! >., n E N, n n . not a measure. t hen f.-l cle=f . f f.-l IS ,
m
n n
2.5.26 Exercise. If I E (Lu n ffi.X ) , for some nonnegative p in Lu and some II in L, 1 = p + II ·
[Hint: If L '3 In t I, 2.5.27 Exercise.
In � 0, while
l in
00
L Un - In- I) E Lu .]
n=2
For some sequence {In } nEN contained d>' t
00 .
2.5.28 Exercise. If En �f [n, (0) , n E N, then: • · •
S ), '3 En ::) En +1 ; . n En = 0 ; nEN >. (En ) == 00 ;
cf. 2.2.26.
m
Ll (ffi., >.) ,
3
Functional Analysis
3.1. Introduction
For a set X, there are various important subsets of CX , e.g., L 1 , L 1 (X, p,), C( X, JR.) , etc. Each of these is an JR.-vector space or a C-vector space and is endowed with a topology related to its manner of definition. Thus L 1 and L 1 ( X, p,) are metric spaces, whereas C ( X, JR.) inherits a topology from CX viewed a Cartesian product. In short, each is a paradigm for a topological as
vector space .
3.1. 1 DEFINITION. A topological vector space (TVS) (V, T) ( OR SIMPLY V) IS A C-VECTOR SPACE ENDOWED WITH A HAUSDORFF TOPOLOGY T SUCH THAT THE MAPS V
x
V '3 (x, y) C x V '3 (z, x)
r-+
x + y E V, r-+ zx E V,
FOR VECTOR ADDITION AND MULTIPLICATION OF VECTORS BY SCALARS ( ELEMENTS OF C ) ARE CONTINUOUS. THE ORIGIN ( THE ADDITIVE IDEN TITY ) OF V IS DENOTED O. WHEN SOME NEIGHBORHOOD BASE FOR T CONSISTS OF CONVEX SETS, V IS A locally convex topological vector space (LCTV S ) . The class of locally convex topological vector spaces includes the class of normed spaces, i.e., the class of those vector spaces V for which there is a norm, namely a map II II V '3 x r-+ Ilxll E [0, (0 ) such that: a) Ilxll = 0 iff x = 0; b) Ilx + yll � Ilxll + Ilyl l ; c ) for z in C and x in V , Ilzxll = Izl . Ilxll . 3.1.2 Exercise. If (V, II I I ) is a normed space, then :
Ilx - yll 2': I llxll - Ilyll l · When ( V, II I I ) is a normed space, d : V 2 '3 (x, y) r-+ Ilx - yll is a metric for V. When (V, d) is complete, V is a Banach space. THEORE M 2.2.32 implies L 1 and, for any measure space (X, S, p,), L 1 (X, p,) are Banach spaces. 89
90
Chapter 3. Functional Analysis
1
When < p < 00, the set LP resp. LP( X, p,) consists of the DLS mea surable resp. S-measurable C-valued functions I such that
1
In Section 3.2 it is shown that if < p < 00 and, according to the convention adopted, each null function is regarded 0, II li p is a true norm and that LP and LP(X, p,) are complete with respect to II li p : each is a Banach space. For a topological space X, CX contains: as
a) Coo (X, q , the set of continuous functions I for which supp (f) is com pact (v. 1.7.23 ) ; b ) Co (X, q consisting of those continuous functions I such that for each positive E, K.(f) �f { x : I I(x) 1 2: E } is compact. For I in Coo (X, q or in Co (X, q , 11111 00 � sup II(x) 1 < 00.
xEX
3.1.3 THEOREM. WITH RESPECT TO II 1 1 00 , Co (X, q IS A BANACH SPACE. PROOF. The verification of the norm properties a) -c ) for II 1 1 00 is straight forward. If {fn }nEN is a Cauchy sequence in Co (X, q , then for each x in X , limoo In (x) exists. {fn (x)} nEN is a Cauchy sequence ( in q , whence I(x) �f n---+ If E > 0, since I ll oo -convergence is uniform convergence, for some N and all x, { m , n > N } ::::} { llm (x) - In (x) 1 < E } , whence
mlim ---+ oo Ilm (x) - In (x) 1
=
II (x) - In (x) 1
� E.
In short, In � I, i.e., nlimoo III - In ll oo o. ( The preceding argument is ---+ valid well if {fn } nEN is a Cauchy sequence in Coo(X, q : for some I, In � I· However, as shown in 3. 1.5, Coo (lR., JR.) is not I ll ao- complete. ) If E > 0, S. de =f { x : I I(x) 1 2: E }, and m 2: 2, then for some nm and all n greater than nrn , Sc C E �f Knm . By defix : I ln (x) l 2: =
as
{
nition, each Knm is compact and S. C
(1 - �) }
n n
Knm � K , which is also
compact. On the other hand, if x E K , m 2: 2, and n 2: nm , then
Section 3.1. Introduction
whence
91
I I(x) 1 2': ( 1 - �) E and x E Sf: Sf
=
K. Thus
I E Co( X, q .
D
[ 3.1.4 Note. When X is compact,
Coo( X, q Co( X, q = C( X, q .] =
� X[ . sink27rx ' n E N, then JR. and In (x) de=f k� ] + k k l , 00 =l . sin 27rX def = I(x) although { In }nEN c Coo (X, q and In (x) -+ � X [k k+ l ] k , k =l I tt Coo(X, q: Coo( X, q need not be a Banach space with respect to the norm I 1 00 ' 3.1.6 Example. The LCTVS Coo(JR., q is a li ll I -dense subset of L l ( JR., >. ) and Coo(JR., q ¥L l (JR., >. ). Hence Coo(JR., q is not li ll I -complete. The following construction, of independent interest, validates the pre ceding statements and provides an explicit I Il l -Cauchy sequence contained in Coo (JR., q and for which the I I I -limit is in L 1 (JR., >. ) \ Coo (JR., q. 3.1.5 Example. If X =
U
""'
For n in N,
In : JR. 3 X f-t nx
n(1 o
x
1
1
if - < < 1-n n - '1f 0 < X < -1
1
x)
-
n 1 if l - - < x < - 1 n
otherwise
Coo(JR., q and In 11�1 X [0, 1 ] . If a < b, there are real constants a , (3 such that if gn (x) �f In (ax + (3) , then gn 11�1 X [ ] . It follows that Coo(JR., q is I Il l -dense in L l (JR., >. ). For a in (0, 1 ) and an enumeration { h h EN of the intervals deleted in the construction of the Cantor set Co: (v. 2.2.40) there are real con-de stants ak, (3k such that if In k(X) =f In (akx + (3k), then supp (Ink) = h 00 and Ink 11�1 X h ' If gn �f Link , then {gn } nEN is a I Il l -Cauchy sequence k=l contained in Coo(JR., q and if its I Il l -limit is g, then X [O , l ] - g X( Ca ) is not a null function and is not in Coo (x, q . is in
a ,b
=
92
Chapter 3. Functional Analysis
3 . 2. The Spaces
LP, 1 � p �
00
Henceforth, L 1 denotes some L 1 (X, p,) or some L 1 derived from a DLS func tional I. As noted earlier, L l (X, p,) and L l are, for appropriately related p, and I, essentially the same. Similarly S and D differ only by a set of null functions. For p in [1, (0), LP { I : I E S, III P E L I } and, when
I E LP, IIIII � = II III P Il l '
�f
3.2.1 LEMMA . ( Young) IF: a) ¢ IS A STRICTLY MONOTONELY INCREASING CONTINUOUS FUNCTION DEFINED ON [0, (0) ; b) ¢(O) = 0; c) '1jJ ¢- 1 ; d)
(x) �
lx ¢(t) dt, AND \II (y) �f lY '1jJ(s) ds;
�f
AND e ) { a, b} C [0, (0) ; THEN ab � (a) + \II (b) . EQUALITY HOLDS IFF b = ¢(a) .
PROOF. In the context, the roles of ¢ and '1jJ resp. and \Ii resp. a and b are symmetric. Hence it may be assumed that ¢( a) � b. The geometry of the situation in Figure 3.2.1 implies that the rectangle [0, a] x [0, b] is contained in
{ (x, y) : x E [0, a] , 0 � y � ¢(x) } U { (x, y) : y E [0, b] , 0 � x � '1jJ(y) } , whence ab � (A) + \II ( b). Equality holds iff b = ¢(a). 1 1 For p in (1, (0), there is in '(I, (0) a unique p' such that - + -
p- The numbers p, p' form a conjugate pair. p' = . p-1 y - axis
I I
I
Figure 3.2.1.
(a,
. 1 2 ; d) The ON
l a>. 1 2 iff for each x in S),
>'E/\ a>. x>. converges to x.
>'E [Hint: For c) , Schwarz's inequality applies to
(L , L ) >'E
(x, x>. )
>'E
(x, x >. ) .
For d), c) applies to prove that n(A) converges to some Y in S) (even if 5 is not maximal) and that x y E 5 1- . For e) , d) and the maximality of 5 apply.] [ 3.2.15 Note. The customary name for S) is Hilbert space. The results c) resp. d) in 3.2.14 are Bessel 's inequality resp. Parseval 's equation. A maximal orthonormal set is often called a complete orthonormal (CON) set. Hence, if {x>.} >'E/\ is a CON and -
v (A ) �f { !(A)
if # (A) < No otherwise
( v is counting measure), S) engenders a measure space
(A, �(A), ) v
so that S) and L 2 (A, ) are isometrically isomorphic.] v
3.2.16 Exercise. If T �f {xn } l < n < N < N o is a linearly independent subset of S), the algorithm represented by the formulre
produces an orthonormal set 5 �f { Yn } l
Section 3.2. The Spaces LP, 1 :::; p :::; 00
b) If
99
X �f {o, l} n �f { y : y �f (Yl " " ' Yn ) } ' Y7 = Yi , S �f �(X), L ° < p < 1, lAy ) �f f.-ln ( Y l , . . . , Yn) = pL:=1 Yk (1 pt - :=1 Yk , f(y ) �f L�=nl Yk , -
then
Ml �f irx f(y ) df.-ln (Y )
1
=
�n
tk ) Yk df.-ln (Y) :2 [tk=l 1 Y% df.-ln + Lk#l 1 YkYI df.-ln (Y)] =
r
X
= � np = p, n
M2 �f x [J(y )] 2 df.-ln (Y ) = x X 1 [ p2 = 2 np + (n2 - n ) p2 ] = ;;P + p2 - --;; , n r [J(y ) - Md 2 df.-ln (Y ) = M2 M� = p(l n- p) :::; � 4n . ix _
Hence
1 0, THERE IS A polynomial function SUCH THAT
IF
f C([O, l],lR) B sup { I f(x) - B(x)1 0 :::; x :::; I } �f I l f - B l oo < E. :
PROOF. There is a positive 15 such that
{I x - y l < r5 } { I f(x) - f(y) 1 < � } . 4 oo If n > sup {r5 -4 , 1 l }, then xk (l - xt- k 1 implies � k=o I f(x) - Bn (f)(x) 1 �f f (X) f ( �) � x k (l - x) n - k '*
��
t( ) l I -� ( ) k (l - x) n - k l ) ( ) f f(x X � �)] ( [ I� =:
=
:::; I L I* - xl ', (3.3.21 ) II T), (z ) 11 :::; l i T), (xn ) 1 1 + II T)' II · ll z - xn l l ·
For large n, the second term in the right member of (3.3.21) is small, whereas the first term in the right member does not exceed 1: l i T), I I :::; � . r When the translation /scaling is reversed, the required assertion follows. D 3.3.22 DEFINITION. A SEQUENCE S �f {Xn} nEN IN A NORMED (VECTOR) SPACE V IS summable IFF summable IFF
00
00
L xn E V.
n=l
THE SEQUENCE S IS absolutely
L I l xn ll < 00.
n=l
Exercise.
3.3.23 a ) A normed vector space V is complete iff every abso lutely summable sequence is summable. b ) The result in a) offers another proof that LP is a complete metric space. [Hint: If: If {xn} nEN is a Cauchy sequence, for some subsequence {Xn k } kEN ' { Xn k+1 - Xnk } kEN is absolutely summable. A modifi cation of the argument in 3.2.7 applies. N
Only if: The partial sums Sn �f L Xn form a Cauchy sequence. ] n=l
3.4. Weak Top ologies
Section
The results in 3.3 deal with the uniform or norm-induced topology for the set [B, F]c of continuous linear operators between the Banach spaces B and F. For some important invpstigations, other topologies are more useful. For a Banach space B, the sequence B, B', (B') '
�f B", . . .
112
Chapter 3. Functional Analysis
is meaningful. For any fixed x in B, (x, x') is a continuous function on B' and thus x may be regarded as an element of B". 3.4.1 Exercise. If B is a Banach space, the map � that identifies each x in B with its correspondent in B" is an injection, and for each x in B, 11�(x) 11 = I l x l l · 3.4.2 Exercise. If B, F are Banach spaces, T E [B, F] and for each x in B, I I T ( x) 11 = I l x l l (T is an isometry), then T E [B, F] c and T (B) is a closed subspace of F. (Hence � (B) is a closed subspace of B".) Owing to the last two results, whenever convenience is served, no dis tinction is drawn between B and �(B). 3.4.3 DEFINITION. THE BANACH SPACE B IS reflexive IFF �(B) = B". 3.4.4 Exercise. The Baill\ch space B is reflexive iff B' is reflexive. (hence, . ). · ff B" , B'" , . . . are refleXlve For an infinit�dimensional Banach space B and its dual B' there are two important topologies different from those induced by their norms. I
3.4.5 DEFINITION. FOR THE DUAL PAIR {B, B'} OF BANACH SPACES, (B, B') resp. (B', B) IS THE WEAKEST TOPOLOGY SUCH THAT EVERY x' resp. x IS CONTINUOUS ON B resp. B'. THESE TOPOLOGIES ARE THE weak ' resp. weak! TOPOLOGIES FOR B resp. B'. THE NOTATIONS BW resp. ( B' ) W ARE USED TO SIGNIFY B resp. B' IN ITS WEAK resp. WEAK ' TOPOLOGY. 3.4.6 Exercise. a) For a Banach space B the set a
a
N (0; x� , . . . , x�; E ) � { x : x:
E B', E > 0, I (x, x; ) I
< E, 1 :::; i :::; n } ,
a
is a convex (B, B')-neighborhood of O. Dually,
a
is a convex (B', B)-neighborhood of 0'. Furthermore, each such neigh borhood is circled, i.e., if ), E C and 1 )' 1 :::; 1, then )'N c N. b) The set NW resp. NW ' of all sucR (B, B')-neighborhoods resp. (B', B)-neighborhoods is a base of neighborhoods at 0 resp. 0'. c) The sets x� , . . . , x� and X l , . . . , Xn may be chosen to be linearly independent without disturbing the conclusions in a) . d) With respect to these topologies B and B' are LCTVSs. 3.4.7 Exercise. For a Banach space B, the weak resp. weak' topology for B resp. B' is weaker than the norm-induced topology. The weak resp. weak' topology is the same as the norm-induced topology iff dim (B) E N. a
a
Section 3.4. Weak Topologies
113 (J"
3.4;8 LEMMA. IF B IS A BANACH SPACE, FOR THE TOPOLOGIES (B, B') AND (B', B), {B, B'} IS A DUAL PAIR AND EACH MEMBER OF THE DUAL PAIR IS THE DUAL OF THE OTHER. PROOF. Since B and �(B) may be regarded as indistinguishable, a) in 3.3.1 is satisfied. If x' E B' and (x, x') 0, then x' 0' by definition. If x i- 0 the Hahn-Banach Theorem (3.3.10) implies that for some x', (x, x') ;f:. 0, whence b) in 3.3.1 is also satisfied. If m E (BW ) ' , since (B, B') is weaker than the norm-induced topol ogy, m is norm-continuous, whence (BW) ' c B'. On the other hand, if x' E B' and E > 0. for any x in N (0; x'; E ) , I (x, x') 1 < E, whence (J"
==
=
(J"
)
' If m E ( (B') W ' there is a weak' neighborhood I,
such that {x . . . , xn } is linearly independent and if y ' E N, then 1m (y') 1 < 1.
( )
rSy, rSy' E N, whence m < 1, 20' 2 0' sup I (Xk , y') I . In particular, if y' E [span (x l , . . . , xn )] i- ,
For any y', if a = sup I (Xk , y' ) I , then l �k�n
i.e., m (y')
' 1 < 1
<E
is circled; for some positive if I r l then r N C N; at 0 there is a base of circled neighborhoods; b) A(x, N) �f { a : 0' 2': 0, x E aN } -j. 0; if N is circled and
PN : V '3 x r-+ inf { a : 0' 2': 0,
x
E aN } ,
then: bI) { {a E A(x, N ) } 1\ {;3 > a}} '* {;3 E A(x, N) } ; b2) for some N, {x -j. O} '* {PN (X) -j. O}; b3) PN (O) = 0 ; b4) for t in C, PN (tX) = I t l pN (X); b5) if N is convex, then PN (X + y ) :::; PN (X) + PN (Y ) ; c) if PN is a function for which bI)-b5) obtains, then { x : PN (X) I } is a convex, circled, absorbent set; d) V is a LCTVS iff for some set {P>. } >. E /\ of functions conforming to b I) � b5), the set N �f { N>. : N>. � { x : P>. (x) 1 } A E A is a base of neighborhoods at 0, and for each v in V \ {O}, there is a A such that
0, for some positive a and ;3 , 0' PN (X) :::; a and ;3 - :::; PN (Y ) :::; ;3 . Furthermore,
E
° and E r5N, then PN ( ) :::; 15, whence m( :::; 15. In short, m is (B", B')-continuous. However, 3.4.8 implies that for some z ' in B', m ( ) = (z ' , ) Since m[�(B)] it follows that z ' = 0', whence m is the zero functional. However, m (y " ) = 1, a contradiction. D
1
u
"
u
- v") 1
- v
a
"
"
u
u
.
u
- v
=
{a},
3.4.12 LEMMA. (Alaoglu) IF B IS A BANACH SPACE, THEN B (0', 1) IS (B', B)-COMPACT. a
I) } [0, I x i Ix. { (x,x') : x' K XXE B lx x' (x') {(x, x')}xEB (O, I ) K x,
x
PROOF. For each in B, E B (0', ]� Ty C is compact in the product topol chonov's Theorem implies �f �f E is, by ogy T. The map (J B (0', 1) '3 r-+ (J virtue of the Hahn-Banach Theorem, injective and (B', B) T continu ous, whence on (J [B (0', 1 )] �f Y, (J- l is T (B', B) continuous. For y in B, and in C, the maps :
a
a
a �x, y : K '3 {aX}xE B r-+ ax +y - ax - ay E C, 1]n ,x : C K '3 (a , {aX}xEB ) r-+ O'ax - anx E C, are continuous. Hence, for each map, the inverse image kx ,y resp. kn ,x of { a } is closed. Thus, [B (0', 1)] = Y (n kx , y ) n (n kn ,x ) is a X ,Y n Ix x
(J
=
a
closed, hence compact set. It follows that B (0', 1) is (B', B)-compact. D 3.4.13 THEOREM. THE BANACH SPACE B IS REFLEXIVE IFF B(O, l) IS WEAKLY COMPACT.
116
Chapter 3. Functional Analysis
]
PROOF. If B is reflexive, then �(B) = B", � IB(O, 1 ) = B (0 " , 1), and the topology inherited by �[B(O, 1)] from the weak' topology of B" is the same as the weak topology of B(O, 1). By virtue of 3.4.12, B(O, l) is weakly compact. Conversely, if B(O, 1) is weakly compact, the (B, B') - (B", B') continuity of � implies �[B(O, 1)] is (B", B')-compact. However, 3.4.11 implies �[B(O, 1)] is (B", B')-dense in B (0 " , 1 ) , whence a
a
�[B(O, 1)
a
a
] = B (0 " , 1) ,
which implies �(B) = B".
D
T
3.4.14 DEFINITION. FOR BANACH SPACES B AND F AND IN [B, Fl c THE ADJOINT IS THE UNIQUE ELEMENT OF [F' , B' e SUCH THAT FOR = EACH IN B F' , [ 3.4.15 Remark. The notation for the adjoint is consistent with the notation for the dual space. Some writers use instead of correspondingly they use instead of
(x, y')
T'
[x, T' (y')] [T (x) , y']. T' V' T*
X
V';
]
T'.]
V*
T' ]
3.4.16 Exercise. a) The statement in 3.4.14 is meaningful, i.e., exists and is unique. b) c) If a , b C C and C [B, F e , then (a s + = as' + Hint: a) The Closed Graph Theorem (3.3.18) applies. b) 3.3.6 applies.]
[
bT)'
I T'I I I T I . bT'.
=
{ }
{S,T}
3.5. B anach Algebras
Gelfand [Gelf] introduced the notion of a normed ring, known today as a Banach algebra. It combines the concepts of Banachology and algebra to form a discipline with many useful developments. Only the outlines of the subject are treated below. Details are available in [Ber, HeR, Loo,
Nai,
Ri) .
Some Banach spaces, e.g. , function algebras such as Co (lR., C ) , form the context for introducing not only addition and scalar multiplication of their elements but also a kind of addition-distributive product of elements. The basic aspects of this development are treated below. 3.5.1 DEFINITION. A BANACH SPACE A THAT IS ALSO A C-algebra IS A Banach algebra IFF FOR AND IN A AND IN C:
a
b z I l ab l :::; I l al l b l ; z(ab ) = (za)b ; I l z al = I z i l al ·
Section 3.5. Banach Algebras
117
Example.
3.5.2 When X is a locally compact Hausdorff space, the Banach 11 space Co (X, q �f A, normed by 00 , is a commutative Banach algebra with respect to pointwise multiplication of its elements; A has a multiplica tive identity iff X is compact, in which case = l .
I
e e 3.5.3 Example. For a Banach space B, the set [B] c cl�f A of continuous endomorphisms (of
B), normed according to the discussion in 3.3.4, is a
Banach algebra with respect composition of its elements; A is commutative iff dim :::; 1; the identity endomorphism id is always the identity for A.
(B)
3.5.4 Exercise. If the Banach algebra A contains a multiplicative identity e such that ea axae11 a: a) e is the only identity; b ) renormed according I to I l x l ' �f sup l 11 , A is again a Banach algebra and I l e l ' = 1; c ) for some #0 l a positive K and all x, K l x l :::; I l x l ' :::; I l x l · Thus I I and I I ' are equivalent norms: If A contains an identity e , I l e l may be taken 1. ==
==
as
3.5.5 DEFINITION. WHEN A BANACH ALGEBRA A CONTAINS AN IDENTITY Ae cle =f A; WHEN A CONTAINS NO IDENTITY, IS A SYMBOL SATISFYING tJ. A AND Ae cle E C, E A } . WHEN A CONTAINS NO =f { + IDENTITY AND {Zie + Xi } i= 1 , 2 C Ae ,
e,
e
ze x : z
x
e
+ Z2e + X2 �f ( Zl + Z2 ) e + ( Xl + X2 ) , ( Z2e + X2 ) � Zl Z2e + ZlX2 + Z2 X l + XIX2 ,
Z l e + Xl (Zl e +
xd
I l ze i I z i I X I .
+X � + AND Ae IS NORMED ACCORDING TO If A = Ae and = is a left inverse of b and b is a right inverse of
ab e, a
a.
Exercise. For a Banach algebra A : a) Ae is a Banach algebra; b) x Oe + x E Ae is an isometry. 3.5.7 Exercise. If A = Ae and and are left and right inverses of x, clef x 1 and every eft Inverse . . . . then (nght mverse ) f x IS X - 1 . 3.5.8 Example. When is counting measure, the classical Hilbert space (v. Section 3.6 ) is L2 (N, v) �f £2 consists of all vectors a �f (al , a2 , . . . ) 00 3.5.6 the map A '3 u = v
=
r-+ -
v
1
of complex numbers such that
u
v
L l an l 2 < 00.
n= l
0
The set [p2 L of continuous
a ter 3. Functional Analysis
Ch p
118
endomorphisms of A is, with respect to composition of endomorphisms as product, a noncommutative Banach algebra. The maps T £2 '3 (al ' a 2 , . . . ) f-t (a 2 ' a3 , . . . ) , S £2 '3 (a I , a 2 , . . . ) f-t a I , a2 ' . . . ) . :
:
(0,
are continuous endomorphisms. Furthermore, T S = id but ST i- id . Thus T[S + (id - ST)] id : both S and S + (id - ST) are different right in verses of T: absent commutativity, right inverses need not be unique. Since S'T' id ' = id and T'S' i- id a similar argument shows left inverses need not be unique. introduces the In Ae , the identity = + expression + If has a right inverse, it may be written as and 0: is meaning = or + + ( ful even when A contains no identity. =
=
y - exy. (ee--x)(x e - y)y e - (x y - xy ) x e - y, (e - x) - y) e x - xy x y - xy =
x
3.5.9 DEFINITION. FOR AND Y ELEMENTS OF A BANACH ALGEBRA A, 0 Y �f X + Y WHEN 0 Y = 0 , Y IS A right adverse OF (AND IS A left adverse OF
x
- xy.y
).
x
x
x
3.5.10 THEOREM. THE BINARY OPERATION 0 IS ASSOCIATIVE. IF U AND ARE LEFT AND RIGHT ADVERSES OF X , THEN U = cle =f X , THE ad verse OF AND PROOF. The associativity of 0 follows by direct calculation. If U 0 0 = 0 , then
V
=
x
°
V
xxo xOx. x x � XO u = u u (x ) = (u x) and u xu - x = ux - x. Hence xOx xux - x2 = xxo. D l 3.5.11 Exercise. a) The adverse XO exists iff (e - X)- exists in Ae. b) 00 If I l x l < 1, then XO exists and XO = - L xn . c) In Ae, n= l (x o y )(e - x) = (e - x)(y ox); =
V
00
=
0
0
v
=
0
=
0
v
=
0 0
v
=
v
d) If XO exists, for any z, 0
0
(z - XC) (e - x) = z x and (e - x) (z - XC) x z. e) If XO and yO exist, then x y is advertible and (x yt = yO xo. f) In Ae , if l i e - x i < 1, then X- I exists. g) In Ae, if X - I exists, for some positive r, y - l exists if I l y - x i < r. 0
=
0
In Ae the set H of invertible elements is nonempty and open.
0
Section 3.5. Banach
Algebras
1 19 00
e + 2:)e - x) n converges and n= 1 [e - (e - x)] (e+ � (e - x) n ) = e. 1 g) If I vi i < -I ' f) implies e + X- I V is invertible.] Ix I [ 3.5.12 Note. The result g) has a counterpart for adverses, v. 3.5.14.] [Hint: f) The series
Exercise.
3.5.13 a) In Ae the set of H of invertible elements is: a group relative to the operation of multiplication in A. b) The map EH H '3 f--t is a bicontinuous bijection, i.e., an auteomorphism.
x X -I
3.5.14 DEFINITION. FOR A BANACH ALGEBRA A, THE SET adv (A) CON SISTS OF THE ADVERTIBLE ELEMENTS, i.e., THOSE FOR WHICH THERE IS A ( UNIQUE ) LEFT AND RIGHT ADVERSE. 3.5.15 LEMMA. FOR A BANACH ALGEBRA A, adv (A) IS AN OPEN SUBSET (cf. 3.5.11g) ) AND ° adv ( A) '3 f--t IS AN AUTEOMORPHISM. :
g gO 1 PROOF. If x E adv (A) and I l y - x i < then 1 + I l xo l ' XO (y - x) - XO(y - x) , y o XO (y - x) - (y - x)XO, xI l o y l � I l y - x i (1 + I l xO I ) < 1, I l y x0 1 � I l y - x i (1 + I l xO I ) < 1, whence XO and y XO are advertible. If is the adverse of XO y then XO is a left adverse of y. If z is the adverse of y xO, then XO ,z is a right adverse of y. Thus y is advertible: adv (A) is open. If x E adv (A) and x + h E adv (A), then I l h - hxo l � I l hl (1 + I l xO I ) , and if I l h l is small, then u �f h - hxo E adv (A). Furthermore, 3.5.11 implies (x + h) o xo = u, (x + ht XO uO, (x + h) ° - XO UO - xO uo , 0Y =
w
0
0Y
0
0
0
=
w
=
=
0
0
0
0
Chapter 3. Functional Analysis
120
I l u l � I l h l ( l + I l x l ! ), when I l h l is small enough, I l u l < 1. Thus I l h l (1 + I l xO I ) 2': I l u l = I l uo - uuo l 2': I l uo l - I l u l I l uo l , I l u° I p and III ( Pj ) - I (aj ) 11 < E, 1 :::; j :::; n. However, (3.5.21) implies that for all n in N, (3.5.22) As in earlier calculations, lim II (a) - n xn ll n --+ =
E
=
0, whence
Because p < a, lim a- n xn = 0, whence lim a - n xn ) O = 0. Since n --+ CXJ n --+ CXJ ( may be arbitrarily small (and positive), by virtue of (3.5.22) ,
whence
lim IIp- n xn li = o. (3.5.23) n --+CXJ However, since p = sr (x) , inf II p- n xn ll -;;- = 1, and (3.5.23) yields a contra n EN diction. D [ 3.5.24 Remark. The derivation above is given in terms the properties of exp as derived in Chapter 1. An alternative proof can be based on Liouville 's Theorem in the theory of holomorphic functions on C, cf. 5.3.29.] 1
3.5.25 LEMMA. IF x IS AN ELEMENT OF A COMMUTATIVE BANACH AL GEBRA A, THEN sr (x) IS sup { I pl : p E sp(x) } �f P(x) . PROOF. a) For some z in sp(x), sr (x) :::; Izl (v. 3.5.19), whence sr (x) :::; P(x) .
Section 3.5. Banach Algebras
123
On the other hand, if w E sp(x) and
Hence, if
y �f w- 1 x, for all large
n
I w l > I z l , then
and some t, 00
I l yn l � :::; t < I, and so
n= l converges to (cf. 3.5.6c) ). Hence w- 1 x is advertible, whereas, since 1 w E sp(x), w- x, is not advertible, a contradiction:
yO
If sr (x)
=?
{{z E sp(x) } 1\ { sr (x) :::; I z l }} { I z l = P (x) } . (1 - E)P(X) < P(x), the argument above shows
=
00
converges and its sum is (z - l x) O , a contradiction since
z E sp(x) :
sr (x) = P(x).
[ 3.5.26 Note. Owing to 3.5.25, the notation sr (x) and the term
D
spectral radius of x for P are justified.]
3.5.27 LEMMA. IF X IS A BANACH SPACE AND Y IS A CLOSED SUBSPACE, THE quotient space XjY ENDOWED WITH THE quotient norm
I IQ
:
e
XjY :1 r-+ inf {
Il xl
: xj Y =
e}
IS A BANACH SPACE. PROOF. That is a true norm is a consequence of the definitions and the elementary properties of inf. If is a Cauchy sequence in Xj Y, for each n , there is an Xn such that xn jY = and + < Thus and so is a Cauchy + < + sequence. If lim X �f x and �f xj Y, then lim = XjY is a Bana�h space. D Thus [ 3.5.28 Note. By definition, if = xj Y, then the map X :1 x r-+ xj Y �f is norm-decreasing.]
I IQ
{en}n EN n. x en e n l I l l I l n T Q n {xn }n EN I l xn - xm l l i en - e>n I Q 2- Tm e n 4� en e: n 4� n e I l e l Q :::; I l x l . e
124
Chapter 3. Functional Analysis
3.5.29 DEFINITION. A left (right) ideal IN AN ALGEBRA A IS A PROPER SUBSPACE R SUCH THAT AR C R (RA C R) . A SUBSPACE THAT IS BOTH A LEFT AND RIGHT IDEAL IS AN ideal. THE IDEAL R IS regular OR mod ular IF THE quotient algebra AIR CONTAINS AN IDENTITY. WHEN R IS A REGULAR IDEAL AND Ul R IS AN IDENTITY IN AIR, IS an identity mod ulo R. CORRESPONDING DEFINITIONS APPLY TO LEFT ( RIGHT ) identities
u
modulo right (left) ideals .
Exercise.
u
3.5.30 For a Banach algebra A: a) If is an identity modulo the (regular ) ideal R, then is an identity modulo every ideal that contains R. b) If R is a regular left (right) ideal, then R is contained in a regular left (right) maximal ideal. c) Every maximal ideal is closed. d) An element has a right adverse iff is not a left identity modulo any regular right maximal ideal. e) If A ¥ Ae a proper subset S of A is a regular maximal ideal iff for some maximal ideal Me in Ae and different from A, S = A n Me. and are in R. b) Some [Hint: a) For every version of Zorn's Lemma applies to the poset of ideals containing R. c) The continuity of multiplication applies. d) If has no right adverse, then R �f E A is a right ideal, tJ. R, and is a left identity modulo R, and b) applies.]
x
u
x
x, ux - x
xu - x
x
{ xy - y : y }
x
x
Example.
3.5.31 In the Banach algebra Co(ffi., q , Coo (ffi., q is a dense ideal contained in no maximal ideal: Coo (ffi., C) is not a regular ideal. 3.5.32 THEOREM. (Gelfand-Mazur) IF A IS A COMMUTATIVE BANACH ALGEBRA AND M IS A MAXIMAL REGULAR IDEAL IN A, THEN AIM IS ISOMORPHIC TO C
I I Q,
PROOF. With respect to the quotient norm AIM is a Banach field (with identity and so if e i- 0, 0 tJ. sp(e). On the other hand, sp(e) i- (/) and if E sp( e), then e - is singular and since AIM is a field, e = The correspondence e r-+ is an isomorphism between AlM and C D Owing to the fact that A r-+ AIM is an algebra-homomorphism, for a commutative Banach algebra A, each regular maximal ideal M may be re garded as a special element of the dual space A': is a multiplicative lin ear functional. Furthermore, if E A, then whence 1.
z
e)
z
ze
x' x
ze.
x' I l e l :::; I l x l ,
I l x' l :::;
3.5.33 DEFINITION. THE SET OF REGULAR MAXIMAL IDEALS IN A COM MUTATIVE BANACH ALGEBRA A IS Sp (A) , THE spectrum of A. The uses of the word spectrum as in spectrum of [ 3.5.34 (when is an element of a Banach algebra A) and spectrum of A can be misleading. However, the distinction between the two usages is clear:
Note. x
x
•
sp(x) is a set of complex numbers;
Section 3.5. Banach Algebras •
125
Sp (A) is a set of regular maximal ideals.]
3.5.35 THEOREM. IF A IS A COMMUTATIVE BANACH ALGEBRA CONTAIN ING AN IDENTITY, Sp (A) , REGARDED AS A SUBSET OF A', IS A WEAK' COMPACT SUBSET OF THE UNIT BALL B (0', 1) IN A'. PROOF. If A 3 A f-t E Sp (A) is a net and (B', B)-converges to some x' in B (0', 1 ) , for each pair {x, y} in A,
M>.
M>. (x, M>. ) (y , M>. ) (xy, M>.) .
(J"
=
The left resp. right member of the equation converges to (x, x') (y, x') resp. (xy, x') , whence x' is a continuous multiplicative linear functional: Sp (A) is weak'-closed. Owing to 3.4.12, B(O', 1) is weak'- compact. D For x in A and in Sp (A) the complex number (x, is denoted and the map ' : A 3 x f-t E C(Sp (A), q is the Gelfand map. The preceding development implies that � 3.5.36 If A is a commutative Banach algebra and E Sp (A) , then E sp(x) . [ 3.5.37 If E sp(x), for some in Sp (A), is an If y tJ. identity modulo then - 1 ) ° and so Hence 2: sup E sp(x) } �f P(x) .
M
x( M )
x
M)
I l x(M) l oo I l x l .
Exercise. M x(M) Note. Z M l Z� l X M. M, fj(M) (z� x(M) x(M) I l x l oo { I pl : p However, if M E Sp (A) and x(M ) w i- 0, then w� l x(M) 1 and W� l X is an identity modulo M. If y �f (W� l X) O exists, then W� l X + w� l xy 0 , =
= z.
=
=
Y -
=
1 + Y - l y = 0,
a contradiction:
W� l X has no adverse, w E sp(x), I w l � P(x) : I l x l oo P(x) . =
In C(Sp (A), q , Sp (A) is to be viewed in its weak' topology. Thus ' may be viewed as a covariant functor from the cate gory of Banach algebras containing an identity (and contin uous ((>homomorphisms ) to the category CF of continuous func tion algebras on compact Hausdorff spaces (and continuous C homomorphisms) [Loo, Ma ] .]
BA.,
c
3.5.38
Exercise.
If (J C f-t C is a C-automorphism, then (J :
=
id .
Chapter 3. Functional Analysis
126
[Hint: If z E
e,
then 8(z) = 8(z · 1 ) .]
3.5.39 Exercise. If A is a commutative Banach algebra, and x E A, then
,
x [Sp (A)] =
{
sp (x ) if A = Ae sp (x ) or sp (x ) \ { O } if A ¥Ae .
Since function algebras of the form C(X, q arise naturally in the study of commutative algebras, the Stone- WeierstrafJ Theorem below takes on added importance. The result is phrased in terms of the notion of a sepa rating set of functions in a function algebra. 3.5.40 DEFINITION. A SUBALGEBRA A OF eX IS separating IFF FOR ANY TWO ELEMENTS a, b OF X, SOME f IN A IS SUCH THAT f(a) i- f(b) ; A IS strictly separating IFF FOR SOME f, f(a) = = 1 - f(b) .
0
3.5.41 Exercise. If A is a commutative Banach algebra,
A, cle=f { X : X E A } is a strictly separating subalgebra of Co (Sp ( A) , q . 3.5.42 Exercise. If A is a strictly separating sub algebra of JR.x , a, b are two elements of X, and { c , d} c JR., then, for some f in A, f (a) = c and f(b) = d. 3.5.43 Exercise. If X is a compact Hausdorff space and A is a II 11 00 closed subalgebra of C(X, JR.) , then A is a vector lattice. [Hint: If f E A, the Weierstrafi Approximation Theorem (3.2.24) implies that I f I is approximable by polynomial functions of f.] 3.5.44 THEOREM. ( Stone-Weierstrafi ) IF X IS A COMPACT HAUSDORFF SPACE, ANY CLOSED STRICTLY SEPARATING SUBALGEBRA A OF C(X, JR.) IS C(X, JR.) . ( EACH STRICTLY SEPARATING SUBALGEBRA OF C(X, JR.) IS 11 1i 00 -DENSE IN C (X , JR.) . ) PROOF. If f E C(X, JR.) and a , b are two points in X , A contains an fa,b such that fa b ( a ) f(a), fa,b (b) = f(b) . If E then ,
> 0,
=
X
Uab �f { x : fab ( ) < f(x) + E } and Vab �f { x
:
X
fab ( )
> f (x) - E }
are open. If b is fixed, {Uab LE x is an open cover of X. Hence there is a fi nite subcover {Ua1b, . . . , Uapb} and, owing to 3.5.43, fb �f inf fap b E A. l �p �P If x E X, for some p, x E Uap b and fb ( ) < f(x) + E. On the other hand, if X
Section 3.5. Banach Algebras
x E Vb
127
p
�f n Vap b, then Ib(X ) > I (x)
- f.
The open cover {Vb h E X admits p =l a finite subcover {Vb" . . . , VbQ } and 3.5.43 implies ¢ �f sup Ibq E A. If l �q�Q x E X, for some q , x E Vbq and I(x) < ¢(x) < I(x) + Eo D [ 3.5.45 Note. a) If A is merely separating, A can fail to be C(X, JR.) , e.g., if X = [0, 1] ' the set A of all polynomial functions that vanish at zero is separating. However, - f
A=
C(X, JR.) n { I : 1(0)
=
O} .
If JR. is replaced by C in the discussion above, the corresponding conclusions are false, v. Chapter 5. What is true and what follows directly from 3.5.44 is that if A is a closed strictly sepa rating subalgebra of C(X, JR.) and 1 E C(X, q , then 1 = u + iv, {u, v} C C(X, JR.) and by abuse of notation, C(X q = A + iA. There is a corresponding statement if A is merely separating.] .
3.5.46 Exercise. If X is a compact Hausdorff space and a subset A of C(X, JR.) is both A-closed and v-closed, the I lloo -closure of A contains each continuous function approximable on every pair of points by a function in A. 3.5.47 Exercise. a) The set
is strictly separating. b) The smallest algebra A over 1I' and containing is the JR.-span of . c) The algebra A + iA is Il l i oo-dense in C(1I', q . d) 2� The map I C(1I', JR.) 3 1 r-+ 1 (eix ) dx is a DLS functional (with an :
1
,
associated measure T ) . e) For the maps W : [0, 27r] 3 x r-+ e ix E 1I', W* : C1I" =
1 r-+ l o W E C [O , 2 �l , I I W* ( f ) 11 2 ' f) The set W* ( (I, 0) ] > 0, then Ed!, O) is a negative set and E> (1, 0) is a positive set. If ! dl-l < 0, then 4.1.9 Exercise. a) Every measurable subset of a positive set is a positive set. (Hence, if El and E2 are positive measurable sets, then El \ E2 and
E
Ix
Section 4.1. Complex Measures
139
EI n E2 are positive measurable sets.) b) If {En} n EN is a sequence of pairwise disjoint positive sets, U nEN En is a positive set. c) If E is a positive set and A E 5, then E n A is a positive set. d) Similar assertions
obtain for negative sets.
4.1.10 THEOREM. (Hahn) IF (X, 5, p,) IS A SIGNED MEASURE SPACE AND
p,(5) C [-00, (0) or p,(5) C (-00, 00] , THEN FOR SOME P IN 5 AND Q �f X \ P AND EACH E IN 5,
P n E, Q n E E 5, p,(P n E) 2': 0, p,(Q n E ) :::; 0, p,(E) = p,(P n E) + p,(Q n E) . PROOF. The argument below is given when p,(5) C [ - 00, (0). A similar argument is valid when p,(5) C (-00, 00]. If 1] �f sup { p,(E) : E 5, and E is a positive set }, then 1] < 00 (v. 4.1.6) . If 1] = 0, then (/) serves for P. If 1] > 0 there is a sequence {En} n EN of measurable positive sets such that p, (En ) t 1]. If P �f U En , then
E
P = EI LJ U �=2 (En \ En- I ) E 5 and for each M in N, 1] 2': p,(P) = p, (EM ) + whence p,(P)
=
00
L
n=M+ I
1]. If E E 5, then
P, (En \ En- d 2': P, (EM ) '
00
p,(P n E) = P, (EI n E) + L p, [(En \ En - d n E] 2': 0 : n= 2 P is a positive set. If Q �f X \ P and Q is not a negative set a contradiction is derived by
the following argument. For some measurable Eo contained in Q, 00 > P, (Eo) > o. But Eo is not a positive set, since otherwise, PLJEo is a positive set and
p, (PLJEo) = p,(P) + p, (Eo) > 1], a contradiction. Hence, for some measurable subset E of Eo, p,(E) < o. In 1
1
1
the sequence -1, - , - - , . . . there is a first, say - - , such that for some 2 3 ml 1 measurable subset EI of Eo, p, (Ed < - - . Then ml -
p, (Eo \ Ed = p, (Eo) - p, (E d > p, (Eo) > o.
140
Chapter 4. More Measure Theory
The argument applied to Eo now applies to Eo \ EI : for some least positive integer m2 and some measurable subset E2 of Eo \ EI , P, (E2 ) < � and m2 of pairwise disjoint sets and least by induction there is a sequence { E } such that positive integers { } -
n nEN
mn nEN
p, ( E2 ) < -
-m12
,
p, [Eo \ (EI U E2 )] = p, (Eo ) - p, ( EI ) - p, (E2 ) > 0 ,
-
00
ex:> 1 < and mk L n= l m n implies that A �f Eo \ U nEN En Hence
measurable subset F of
then F U EK c Eo \
-+ 00
as k -+ 00 The earlier argument .
is not a positive set. Hence, for some
1 A and some mK , m K > 2 and p, ( F ) < - -m K . But
(u ::11 ) and Ek
mK :
imply a contradiction of the minimality of Q is a negative set and the pair {P, Q} performs as asserted. 0 4.1.11 Remark. The pair (P, Q) is a Hahn decomposition of X. [ Although as constructed, P if X tJ- 5, then Q tJ- 5. ]
E.S,
4.1.12 Exercise. If (Pi , Qi ) , i = 1, 2, are Hahn decompositions of X for the signed measure space (X, 5, p,) , then for any measurable set
(A n PI ) = p, (A n P2 ) (and hence p, ( A n Q I ) = p, (A n Q 2 ) ) '
A,
p,
[ 4.1.13 Note. If p,(N) = 0 and N \ P -j. 0, then a second Hahn decomposition is (P U N, Q \ N). Examples of such N abound. For example, if f �f X [0, l and p, in (JR., 5)" p,) is defined by
1
p, (E) �f Ie f dx,
Section 4.1. Complex Measures
141
then P �f [0, 1] is a positive set and Q �f ffi. \ { [O, I] } is a neg ative set and (P, Q) is a Hahn decomposition. If N �f Q, then IAN ) = 0, N \ P -j. 0, and (P U Q, Q \ Q) is a second Hahn de composition. Thus Hahn decompositions are not necessarily u nique but they all produce the same effects.] 4.1.14 DEFINITION. WHEN (X, 5, f.-l) IS A SIGNED MEASURE SPACE AND (P, Q) IS A HAHN DECOMPOSITION FOR f.-l, THEN FOR E IN 5, f.-l +
( E) clef f.-l (E n P), f.-l- ( E) cle=f -f.-l ( E n Q). =
[ 4.1.15 Remark. In light of 4.1.12, the set functions f.-l ± are independent of the choice of (P, Q).] 4.1.16 Exercise. The set functions f.-l± are (nonnegative) measures. 4.1.17 THEOREM. IF (X, 5, �) IS A COMPLEX MEASURE SPACE, THE SET FUNCTION
I�I : 5 3 E r-+ sup
{�
I� (En)1 : {En} nE N a measurable partition of E
}
IS A MEASURE AND 1�1 (5) C [0, (0 ) PROOF. Since the only measurable partition of 0 is itself, 1�1 (0) = 0. If { En} n EN is a measurable partition of E, then for each n in N and each positive E, En admits a measurable partition {Enk hEN such that 00
'" � I� (Enk ) 1 � I�I (En ) k=1
.
- 2: ' Since E
=
U {k, n }CN Enk , it follows that 00
{k, n }CN whence IWE) �
00
L I�I. (En ) :
n =1
I�I is superadditive.
On the other hand, the partition {En} n EN may be chosen so that 00
n= 1
00
n= 1
142
Chapter 4. More Measure Theory
whence I�I is countably additive: I�I is a measure. The set functions a �f � [�] , �f �[�l are signed measures and
(3
0'(5) U (3 ( 5 ) c ffi..
o'±
(3±
Corresponding to the four measures and there is the Jordan decom 0'+ - 0'_ + i of �. Since the ranges of the measures are contained in [0, (0), it follows that for E in 5 and in the context above,
position �
o'±, (3±
((3+ - (3- )
=
00
00
n=1
n=1
The definitions of
M �f sup {
o'±, (3± imply that + ( D ) + _ ( D ) + (3+ (D) + (3- (D) D E 5 } < 00 ,
O'
O'
whence IWE) :::; M. 4.1.18 Exercise. If (X, 5, f.-l) is a signed measure space, then
D
f.-l f.-l+ - f.-l - and If.-l l = f.-l+ + f.-l - . =
4.1.19 DEFINITION. IF (X, 5, �) IS A COMPLEX MEASURE SPACE,
L l ( X, � ) clef L 1 (X, o'± ) n L =
1
(X, (3± ) ,
4.2. Comparison of Measures
When (X, 5, f.-l) and (X, 5, �) are measure spaces or complex measure spaces the relation between f.-l(E) and �(E) as E varies in 5 deserves analysis. Ex amples of such a relation are: a) 0 :::; f.-l(E) + �(E) < 00; b) I WE) :::; f.-l(E) ; c) (more generally) Ll (X, O C L 1 (X, f.-l). Although each of the preceding is of some interest, the relations that have emerged as of fundamental im portance are those given next .
Section 4.2. Comparison of Measures
143
(X, 5, I-l) AND (X, 5, �), I-l IS � (I-l « 0 IFF {�(E) = O} {1-l(E) = O} ; W HEN (X, 5, I-l) AND (X, 5 , 0 ARE COMPLEX OR SIGNED MEASURE SPACES, I-l � IFF { I WE) = O} {1-l(E) = O}. WHEN AIL E 5 AND I-l(E) I-l (E n AIL ) , I-l lives ON AIL' WHEN I-l LIVES ON AIL ' � LIVES ON A� , AND AIL n A� = 0, I-l AND � ARE mutually singular (I-l -1 �). 4.2.2 Example. If (X, 5, I-l) is a measure space, I is nonnegative and 5measurable, and, for each E in 5, �(E) �f Ie I dl-l , then � « I-l. 4.2.3 Example. For the map I-la : 5), E A (E n Ca ) , ([0, 1], 5)" I-la) is a measure space and I-la lives on Ca. If = 0, then A lives on [0 , 1 ] \ Ca and I-lo -1 A. Although the circumstances just described exemplify the relations � « I-l and � -1 I-l, 4.2.1 DEFINITION. FOR MEASURE SPACES absolutely continuous WITH RESPECT TO =?
�
=?
==
3
a
r-+
the following discussion provides a more refined sorting out of the possibil ities for those relations.
(X, 5, I-l)
(X, 5,�)
I-la
I-ls: I-l = I-la I-lsi
4.2.4 THEOREM. IF AND ARE TOTALLY FINITE MEA SURE SPACES, THEN FOR SOME MEASURES AND a) b) + c) -1 d) -1 e) FOR SOME NONNEGATIVE INTEGRABLE AND EACH IN
I-la « I-l; I-ls I-l; I-la I-ls ; h, E 5, l-la(E) = Ie h d�.
PROOF. (von Neumann) The following argument derives a)-e) more or less simultaneously. If �f + and then Owing n to Schwarz's inequality (3.2.11),
p I-l �
I E L2 (X, p),
I E L2 (X, I-l) L2 (X, �). 1
I d � i I I I dl-l � i I I I dp � (i 111 2 dP) [p(X) ] � < 00. i l-l I l Hence T : L 2 (X, p) I T(f) �f i I dl-l E C is in [L2 (X, p ) ] ' . Riesz 's result (3.6. 1) implies that L 2 (X, p ) contains a g such that for every E in 5 and every I in L 2 (X, p), (4.2.5) Ie dl-l = i XE dl-l = L XEgdp = Ie gdp, L ( l - g ) ldl-l = i lg dP - i lgdl-l = i lgd�. (4.2.6) "2
3
r-+
•
Chapter 4. More Measure Theory
144
If p(E) > 0, (4.2.5) and the inequality 0 :::; f.-l(E) :::; p(E) imply 0 :::;
1)] = 0: 0 :::; g(x) :::; 1 a.e. Modulo a null set l } l:J { x g(x) = 1 } �f Al:JS . Thus, for E in
Thus g ? 0 a.e. and p [E> (g, (p) , X = 0 :::; g(x) < 5,
{x
� · Ie g dp :::; 1.
p( )
:
f.-l(E n
:
A) + f.-l(E n S) �f f.-la (E) + f.-l s (E), f.-l a « f.-l, f.-ls -1 f.-l.
For n in N, E in 5, and f �f
(� g k ) . X E ,
(4.2.6) reads (4.2.7)
A,
As n -+ 00, if E C the left member of (4.2.7) converges (by virtue of Lebesgue's Monotone Convergence Theorem) to f.-la(E) and the integrand in the right member of (4.2.7) converges monotonely to some h: for any E in 5, f.-la(E) h d�. In particular, 0 :::; h E L l (X, � ) . D ==
Ie
[ 4.2.8 Remark. The equation f.-l = f.-l a + f.-ls represents Lebesgue 's decomposition of f.-l: f.-l a is the absolutely continuous component and f.-ls is the singular component of f.-l. The equation f.-la(E)
==
Ie h d�
is the expression of the Radon-Nikodym Theorem . The complex of results and assertions is sometimes referred to as the Lebesgue Radon-Nikodym Theorem or LRN. The function h is the RadonNikodym derivative of f.-l with respect to �: h = df.-l .
d�
]
4.2.9 Exercise. The Radon-Nikodym derivative of f.-l is unique modulo a
null function (f.-l).
4.2.10 Exercise. The validity of LRN persists if f.-l is a complex measure and if X is totally a-finite (with respect to If.-ll). 4.2.11 Exercise. If (X, 5, f.-l) and (X, 5 , �) are measure spaces such that X E 5 and f.-l(X) + �(X) < 00 ((X, 5, f.-l) and (X, 5, O are totally finite) ,
Section 4.2. Comparison of Measures
145
� « p" and � is not identically zero, then for some positive E and some E in 5, E is a positive set for � - Ep,. [Hint: For each n in N, if (Pn , Qn ) is a Hahn decomposition for � - p" then � Qn ° < � U Pn . For some no, n EN n EN 1 p, (Pno ) > 0, E = Pno , and E - .] no
(n )
�
=
(
)
=
4.2.12 Exercise. If X in (X, 5, p,) is totally finite and � « p" then : a)
{
}
r f dP, :::; �(E) -I 0; A �f f : O :::; f E L l (X, p,), M(E) �f sup E E S JE
b) for some nonnegative h in S and each E in 5,
l
��).
l h dv
=
M(E); c)
�(E) == h dp, (hence h (The preceding conclusions yield a second proof of LRN.) [Hint: For c), 4.2 . 11 applies.] d� = h, and is in 4.2.13 Exercise. If (X, 5, p,) is totally finite, � « p" g dp, LOO (X, �), then 9 d� 9 h dP,. =
1
=
1
4.2.14 THEOREM. I F (X, 5, p,) IS A MEASURE SPACE AND (X, 5, O IS A COMPLEX MEASURE SPACE, THEN � « p, IFF FOR SOME MAP
J : (0, (0 )
3
THE IMPLICATION {p,(E) < J(E)}
E r-+ J(E) E (0, (0 ) ,
::::}
{ I�(E) I < E } OBTAINS. PROOF. If � « p, and no J as described exists, then for some positive E, each n in N, and some En in 5, p, (En) < T n while I� (En ) 1 2': E. Then p, lim En = ° while I�I lim En 2': E, a contradiction. n 4� n 4� If J as described exists, p,(E) 0, and E > 0, then 1p,(E)1 < J(E), and thus I�(E)I = 0. D 1 4.2.15 Exercise. If (X, 5, p,) is a measure space, h E L (X, p,), and for E in 5, �(E) �f h dp" then = h and IWE) l h l dp,. [ 4.2.16 Remark. In the preceding discussion there are references to various special measure spaces, e.g., a (X, 5, p,) that is totally finite.
(
)
(
l
��
=
)
=
l
146
Chapter
4.
More Measure Theory
For a given measure space (X, 5, p,), the following classification of possibilities is useful. A measure space in one class belongs to all the succeeding classes. a) (X, 5, p,) is totally finite, i.e. , X E 5 and p,(X) < 00; b) (X, 5, p,) is finite, i.e. , for each E in 5, p,( E ) < 00; c) (X, 5, p,) is totally a-finite, i.e., X is the union of count ably many measurable sets, each of finite measure; d) (X, 5, p,) is a-finite, i.e. , each E in 5 is the union of countably many measurable sets, each of finite measure; e) (X, 5, p,) is decomposable, i.e., 5 contains a set F of pairwise disjoint elements F of such that: e1) X = U F; for each
FE F F in F, p,(F) is finite; e3) if p,(E ) is finite, p,(E ) =
L p,(
FE F
E
n F),
e2 )
27
( 4. . 1 )
(whence there are at most count ably many nonzero terms in the right member of e4) if A C X and for each F in F, A n F E 5, then A E 5. A set E in 5 is finite, a-finite or decomposable according as, by abuse of notation, (E, E n 5, It ) is totally finite, totally a-finite, or decomposable. Discussions of the relations of the hierarchy to the validity of LRN (and hence the validity of the representation theorems in Section 4.3) can be found in [GeO, Halm, HeS, Loo] .]
(4. 2. 1 7));
4.2.18 Exercise. If f E
L 1 (X, p,), then the set K", (f, O) is a-finite.
4.2. 19 THEOREM. IF (X, d) IS A METRIC SPACE, (X, aR[K(X)], p,) IS FI NITE, AND EACH x IN X IS THE CENTER OF A a-compact OPEN BALL, THEN (X, aR[K(X)] , p,) IS REGULAR.
PROOF. The set R of regular Borel sets is nonempty since (/) E R. The formulre of set algebra imply that R is a ring. If E > 0, R 3 An C An+1 C Un+1 E O(X), and p, (Un \ An ) < 2En ' then 00
A �f U An C U Un �f U E O(X) , nEN n=2 00 00 00 U \ A C U (Un \ An ) , p,(U \ A) � L p, (Un \ An ) < L 2: < E. n=2 n=2 n=2
Section 4.3. LRN and Functional analysis
147
N
U An �f BN , then BN E R. Lebesgue's Monotone Convergence Theon= 1 rem and the finiteness of (X, a R[K (X)], I-l) imply that for large N, If
Furthermore, B N contains a compact N such that BN ) + (BN N ) E : Hence N If E > 0, 3 :J and :J
K I-l (BN \ KN ) < 2E ' I-l (A \ K ) .::;: I-l (A \ I-l \ K < A E R. R Dn Dn+1 Kn+1 E K (X), I-l (Dn \ KNn ) < E, then D �f n Dn :J n Kn �f K E K(X) and I-l(D \ K) < E. If n Dn �f EN , n= 1 nEN n=2 for large N, EN E R, (EN \ D) < �, and EN is contained in an open set UN such that I-l (UN \ EN ) < 2E ' Hence =
It
D E R.
R
R < < < < .!.
Thus is monotone, whence is a a-ring (v. 2.5. 11) . Lebesgue's Monotone Convergence Theorem implies that a a-compact open ball B ( x , t is regular. If 0 s then B ( x , s t is also regular. If then for each n in N and each k in there is a regular open ball B k t such that 0 and so for some finite set {kih < i 0, and for some n, .!. < 15: tJ- Ur" i.e., n K = n Un . Since R is a a-ring, K is regular: a R[K(X)] C R. D nEN K, K
rk,
x
4.3. LRN and Functional analysis
Among the important consequences of LRN are the characterizations of the dual spaces and when X is a locally compact Hausdorff In particular, the problem raised at the beginning of space, the Chapter can be addressed. It is no exaggeration to state that modern functional analysis owes its current richness to the role played by LRN in establishing the basic relations among what are now regarded as the classical function spaces.
[LP (X, I-l)] ' , [Co(X, C)] ' .
Chapter 4. More Measure Theory
148
4.3.1 THEOREM. IF (X, S , p,) IS TOTALLY FINITE, 1 '::;: p < 00, AND L E [U(X, p,)]' ,
Ix
THEN FOR SOME f IN U' (X, p,) , L( g ) == gf dp,. THE FUNCTION f IS UNIQUE (MODULO A NULL FUNCTION) AND Il f ll p ' = I I L I I . [ 4.3.2 Remark. The result above is in one sense a generaliza tion of 3.6.1; the assumption that (X, S, p,) is totally finite limits the generality. On the other hand, extensions to totally a-finite measure spaces are available (v. 4.3.4) . 1 PROOF. At most one such f exists since for any set E in S ,
The heart of the argument centers on showing that the ([-valued set function � S 3 E r-+ L is a complex ,measure, and that � « p,. The dp, serves for f . complex conjugate of Radon-Nikodym derivative d� The reasons that � is a complex measure and � « p, are: a) :
(X E )
b) L is linear; c) L is continuous (whence for each g in LP (X, p,) ,
Because X is finite, LOO (X, p,) C U(X, p,) . If g E LOO (X, p,) , there is a sequence s n nEN of simple functions such that
{ }
Sn t g
ll ..(. 0
and I lg - s n oo
(v. Section 2.1), whence n---+ limoo Ilg - sn ll p
=
O. Thus, if
then (4.3.3) (v. 4.2.13). The next paragraphs show: f E U' (X, p,) and I f l l p'
=
IILII.
Section 4.3. LRN and Functional analysis
149
l ie l
If p = 1 and E E 5 , then I dl-l .::;: II L II . I-l(E) , whence as in the PROOF of LRN, it follows that II (x) 1 .::;: II L II a.e.:
I E L00 (X, I-l) [= £P' (X, I-l)] .
If 1 < p < 00, then sgn (7) is measurable and sgn (7) 7 = If En �f E:;(II I , n), n E N, and kn �f x En lll pl - 1 , then
III (2.4.8).
I kn l P = xEn lllpl and kn E LOO (X, I-l ). Hence
1En Illpl dl-l = Jx
r kn · k�- 1 dl-l
Ix
Ix
= kn lll = kn sgn (7) 7 dl-l = L [kn sgn (7)] � II L II · llkn sgn (7) li p � II L ll ll kn il p
)
1
)
1
and Il kn ll p P implies pr � l i L l i, n E N. = n Thus Il h ll p .::;: II L II . Holder's inequality and (4.3.3) imply II L II .::;: 11 1 11 pl . Thus II L II =l Il h ll p , 1 � P < 00. D l 4.3.4 Exercise. The conclusion of 4.3.1 holds if (X, 5 , I-l) is totally a-finite and 1 '::;: p < 00: [£p (X, I-l)]' = £P' (X, I-l). [Hint: If X = U nEN Xn , Xn E 5 , and 0 < I-l (Xn ) < 00, by abuse . of notatlon, there are 5 n cle=f 5 n Xn, I-ln cle=f I-l I Sn ' ( Xn , 5 n , I-ln ) , B anach spaces Bn �f LP (Xn , I-ln ), injections
(le
Ill pl dl-l
(Ix
x En lll pl dl-l
that identify LP (Xn , I-ln ) with closed subspaces of LP(X, I-l), and finally Ln �f L I Bn For each n there is in Lpl (Xn , I-ln ) an hn such that for g in Bn , Ln ( g ) = r g hn dl-ln and '
JXn
Each I in LP(X, I-l) may be written uniquely in form
00
00
L I IXn �f L ln' n 1 n 1 =
=
150
Chapter 4. More Measure Theory
00 n=1
The result 2.3.2 can be extended to
X Co (X, q f 1 (1 )
4.3.5 THEOREM. (F. Riesz) IF IS A LOCALLY COMPACT HAUSDORFF THERE IS A SPACE AND 1 : 3 r-+ E e lS IN REGULAR COMPLEX MEASURE SPACE p, ) SUCH THAT
(X, S,B, 1 (1 ) Ix f(x) dp,.
[Co(X, q]"
=
�[1(1 )] � 0' (1) 'S[1(1)] (3(1), a (3) a
(3
then and are continuous PROOF. If , �f and linear JR.-valued maps, by abuse of language, signed functionals. The ar gument reduces to showing that (and similarly is further decomposable into the difference of (nonnegative) DLS functionals to which 2.3.2 applies. There is an echo in what follows of the Hahn decomposition (4.1. 10) of a signed measure. E JR.) �f sup For in and = 2: Thus sup is abbreviated If c 2:
f cto(X, , 0'+ ( 1 ) + { la(g) 1 : g Coo+ (X, q, I g l �f foo.} la (g) l . 0' (1 ) 0' (0) 0 0' (1 ) � I l al l l ll I gl � f
0, then
Igi I fi , i 0'+ (II ) + 0'+ (h ) =
Moreover, if = 1, 2, then the careful application of the identity � sgn (z)z == Izl leads to
10' (gI )l + I g2sup1�h 10' (g2 ) 1 sup [ 10' (g I ) l + 10' (g2 )1 ] 1911Sh 1 92 I S h = sup a {sgn [a (g I )] g l + sgn [a (g 2 )] g2 } 1911Sh 192 1 S h 10' (g l + g2 ) 1 � 0'+ (h + h ) · .::;: 191sup 1Sh sup
Ig l l �h
=
192 I S h
I g l .::;: h 12
0 � h 1\ I g l �f hI .::;: h , 0 .::;: Igl - h I �f h2 � h Fur gI l h I + h2 ,
If + , then thermore, since =
whence
0' + (h + h ) � 0'+ (h ) + 0'+ ( h ) · If f E Coo(X,JR.) and
Section 4.4. Product Measures
151
0' - (1+ ) 2: 0,
0'+- (1 ) - 0'+ (1) - 0' (1 ), 0' - 0' .
0'+ (1 ) 2:
then �f then max{O, a(l)} . If Similarly analysis applies to (3. a ± are continuous, and a = Via the DLS procedure, the functionals a±, (3± engender regular measures (± , 1]± . - 1] - ) and E Coo(X, and E Coo (X, If f.-l �f - C + i
11
then ( )
=
(+
(1] +
r f(x) df.-l, and �
ness of Coo (X,
f
1 f.-l I (E). 11 1 11 = Esup E�
q in Co (X, q applies.
q
f
q,
Finally, the I ll oo - dense-
D
4.4. Pro duct Measures
To measure spaces (Xi, 5i, f.-li) , i = 1, 2, there correspond: a) the space X X l X2 and b) the intersection 5 1 8 5 2 of all a-rings contained in lfj (Xl X2 ) and containing { E l x E2 : Ei E 5i, i = 1 , 2 }. There arises the question of how to define a product measure f.-l on 5 1 8 5 2 so that the equation f.-l (E l x E2 ) = f.-l l (Ed ' f.-l 2 (E2 ) is satisfied for all Ei in 5i, i = 1 , 2. The DLS approach to the answer is given below. 4.4.1 Remark. Alternative derivations can be given by prov ing whatever is claimed first for simple 5 1 8 5 2 -measurable func tions and then, via the approximation and limit theorems of mea sure/integration, extending the conclusions for wider classes of functions. In such a procedure, the result---a pre-Fubinate mea sure space (X l x X2 , 5 1 8 5 2 , f.-l l 8 f.-l 2 )-can fail to be complete.] Associated with (Xi, 5i, f.-li) are the Banach spaces L l (Xi, f.-li) , i = 1 , 2, and the nonnegative linear functionals Ii : L l (Xi , f.-li ) 3 f-t r df.-li, i = 1, 2. }xi If Ap , 1 :::; p :::; P, resp. Bq , 1 :::; q :::; Q, in 5 1 resp. 5 2 are sets of finite mea sure (f.-ld resp. (f.-l 2 ) and apq , 1 :::; p :::; P, 1 :::; q :::; Q, are real numbers, then P,Q �f "'" � apq X A p X Bq p,q=l is in ffi.x 1 X X2 and the set L of all such functions is a function lattice.
�
X
X
[
f
f
f
P, Q 4.4.2 LEMMA. a) IF � "'" apqX Ap X Bq E L, THEN 5 1 resp. 5 2 CONTAINS p,q=l PAIRWISE DISJOINT SETS Eu , 1 :::; u :::; resp. Fv, 1 :::; v :::; V, AND ffi. CONTAINS NUMBERS auv , 1 :::; u :::; 1 :::; v :::; V, SUCH THAT P,Q L apq X Ap XBq = L auv X Eu X Fv ' u , v=l p,q=l
U,
U,
u,v
152
Chapter 4. More Measure Theory
b) FURTHERMORE, P, Q
U, V
p,q=l
u , v=l
L apq h (X AJ 12 (X BJ = L
( J h (X FJ ·
O'uvh X E
PROOF. a) A detailed computational argument establishes the validity of the assertions in the LEMMA. Not only is the argument tedious, it is not really informative and adds little to the understanding of the underlying structure shown in the visible geometry. The basic reasoning is depicted in Figure 4.4.1. The interested reader might wish to provide a formal argument that translates the geometry into the unavoidable prolixity. b) The linearity of the functionals h and h implies the result. D P, Q 4.4.3 Exercise. If L apq X Ap X Bq cle =f f E L, then p,q=l
is well-defined, i.e.,
1( 1 ) is independent of the representation of f.
UI x V3
U2 x V3
U3 x V3
UI x V2
U2 x V2
U3 x V2
UI x VI
U2 x VI
U3 x VI
L---�----�-- X I U2 Figure 4.4. 1.
Section 4.4. Product Measures
153 X
4.4.4 DEFINITION. WHEN A C Xl X2 AND X E Xl , THE x-section Ax OF A IS { X2 : (x, X2) E A }. WHEN f E ffi.X 1 X X2 THE x-section fx OF f IS X2 : 3 X2 r-+ f (x, X2) . SIMILAR FORMULATIONS APPLY FOR Y IN X2 , Ay , AND fy . 4.4.5 LEMMA. THE MAP I IS A DLS FUNCTIONAL. PROOF. The nonnegativity and linearity of I flow from its definition. Fur
P,Q p,q= l
thermore, if f �f " � apq X Ap X Bq E L, then for each ( fixed ) x in X l ,
fx
=
12 (fx ) =
P,Q L apq X Ap (x)XBq E L I (X2 ' f.-l2) , p,q= l P,Q L apq X Ap (x) h (X BJ E L l (Xl , f.-l d · , q=l
p
Similar formulre apply for fy . The definitions of the functions imply the fundamental equality: 1(f) = II [12 (fx )] h [h (fy )]. Because (fn ) x ..!. h [(fn U ..!. O. Furthermore,
0,
=
L l (Xl , f.-l d 3 h [(fn ) x ] ..!. 0,
whence h {h [(fn U } = I (fn) ..!. o. D Since I in 4.4.5 is a DLS functional, in accordance with the develop ments in Chapter 2, I engenders a complete measure space (X, 5, f.-l) , the Fubinate of (X l , 5 1 , f.-l d and (X2 ' 52, f.-l2) :
When the number of ingredient factor spaces is two or more, the gen eral vocabulary and notation for dealing with product measure spaces are those given in 4.4.6 DEFINITION. a) FOR A SEQUENCE { (Xn , 5 n , f.-ln ) } nEN OF MEASURE SPACES, : (Xl x X2, 5 1 52, f.-l l x f.-l2) IS THE Fubinate OF (Xl , 5 1 , f.-l d AND (X2' 52, f.-l2) ' b ) FOR GREATER THAN 2,
K
X
-l K-l K K-l X 5 , , k k X X X f.-lk)
IS THE Fubinate OF
(
k=l
k=l
k= l
154
Chapter 4. More Measure Theory
X k= l k , THE INTERSECTION OF ALL a-RINGS CONTAINED IN !fj(Y) AND CONTAINING { X �= Ek : Ek E Sk, 1 ::; k ::; K } IS 6Sk. l k= l ON OSk, Xk f.-l k IS A MEASURE DENOTED Of.-lk AND THERE EMERGES k=l =l k=l THE pre-Fubinate MEASURE SPACE X �= Xk, g Sk, g f.-lk . WHEN ( ) l EACH Si == 5, AND EACH f.-l f.-l , THE FORMULJE f FOR Y cle = K
Xi
==
K
X
K
K
X,
i
==
f.-l) ,S,f.-l) 4.4.7 Exercise. a ) If K 2: 2, then Xk Sk :J OSk. b ) If Ek E Sk, then =l k=l Ek) fl f.-ldEk) ' (X �From ( X �=l 4.5.16, =l f.-lk )4.5.10, and 4.5.18 below it follows that some stances :J in 4.4.7a) is and in others :J is -;;. .
PROVIDE THE NOTATIONS FOR THE K-FOLD FUBINATE OF ( X WITH ITSELF AND THE K-FOLD PRE-FuBINATE OF (X, 5, WITH ITSELF. K
K
=
m
=
m-
Fubinate measure spaces arise from DLS functionals and are automatically complete. Some pre-Fubinate measure spaces can fail to be complete.
The procedure described in 4.4.1 leads to pre-Fubinate measure spaces. Their completions are the Fubinate measure spaces. One advantage of the DLS approach to product measures is the fact that it immediately produces Fubinate measure spaces.
f.-l) )
4.4.8 Exercise. a) If (X, 5 , is a measure space and 5 contains a ring R of sets of finite measure, then a R ( R is a-finite. b ) The measure space (X, 5, engendered by I in 4.4.5 is a-finite. [Hint: As in 2.2.40, for a) the relevant sets are Eo, the set of all countable unions of elements of R, and for each ordinal number a in (0, Q) , the set En of unions of count ably many set differences drawn from U
f.-l)
-y.. , X), E and b) I-l)' (X),) = 1 are imposed.
{S)' hEA S ),
),E),
{ 1-l), hEA
The customary approach is to form, for each finite subset
the rectangle R �f E)' l . . . x E), x X X)" then the algebra ), $ a F, and the a-algebra A generated by all such rectangles. For each X
n
n
R, I-l( R) �f II I-l), k (E)' k )' The set function I-l is extensible to a k= l
measure on A and this measure behaves properly with respect to of given measures. the set
{ 1-l), hEA
Details of this procedure and of the associated Fubinoid theorems can be found in [HeS] .]
4 . 5 . Nonmeasurable Sets
The existence of nonmeasurable sets in the context of a measure space (X, I-l) can be established trivially in some instances and in others, only appeal to sophisticated aspects of set theory permits a satisfactory resolu tion.
S,
R �f for I-l : R 3 E f-t 0, (X, R , I-l) is a complete measure space and every nonempty subset of X is nonmea surable. b) The set C consisting of and all finite or countable subsets of JR. is a a-ring and C 3 E f-t #(E) is counting measure. If X �f JR., the measure space (X, C, ) is complete, and every noncountable subset of X, e.g., JR., is nonmeasurable. (Note that if E E R resp. E E C and M e X, then M n E E R resp. M n E E C.) By contrast, the following discussion demonstrates that for the com ).. ) , there is a set M such that if )"(E) > 0, then plete measure space (JR.,
4.5.1 Example. a) If X i- (/) and v :
{(/)},
(/)
v
S)"
Section 4.5. Nonmeasurable Sets
159
M n E tJ- 5), . The result is applied to the study of the completeness of product measure spaces that arise from complete measure spaces. 4.5.2 Example. The complete a-algebra 5 generated by the set of all arcs A:> ,(3 �f { : 0 ':::; a < (J < (3 .:::; 27r } in 'lI' may be endowed with the mea sure 7 (v. 3.5.47) such that 7 ( A ,(3 ) �f (3 - a. If (J E JR., then Aa,(3 E 5 and 7 ( Aa,(3 ) = (3 - a: ('lI', 5, 7) is In the group 'lI' there is the subgroup �f { : (J E Q } . The Ax iom of Choice implies the existence in 'lI' of a set 5 meeting each coset of in precisely one element. Thus { g5 : 9 E consists of count ably many pairwise disjoint sets such that U E g5 = 'lI'. If 5 E 5 and 7(5) = then g G for 9 in 7(g5) = for each in N, 7('lI') 2: and so = O. Since 'lI' = U E g5, it follows that 7('lI') 0, a contradiction. Thus 5 C!fj('lI') . g G 'f=The map exp : JR. 3 r-+ E 'lI' (v. 2.3.13) permits a transfer of the dis cussion above from ('lI', 5, 7) to (JR., 5)" A). The conclusions reached are paraphrased loosely by the statement: 'lI' and JR. contain nonmeasurable sets. 4.5.3 Exercise. a) For disjoint sets E l and E2,
e iO e'O G,
a translation-invariant. e iO G e iO G G} a, n na, a
a:
=
x e ix
(4.5.4 )
b) for each F in 5 and E in H (5), p, * (F n E) + p,* (F \ E) = p,(F) ; c) if E E H (5) and p,* ( E ) < 00, then E E 5 iff p,* ( E ) p,* ( E ) . a) For the first inequality in (4.5.4) there are sequences {An} nE N resp. {Bn} n E N contained in 5 and such that =
[Hint:
An C El and P, (An) t p,* (E d , Bn :J E2 and p, (Bn ) ..(. p,* (E2) ;
for the second inequality, there is an 5-sequence {en} n E N such en :J en+ l and p, (en) ..(. p,* (El l:!E2 ) . Then p, (An n en) t p, * (E d and P, (Bn n en) ..(. p,* (E - 2). For b) , a) applies.] 4.5.5 Example. If a is an irrational real number, � �f A
�f { C : n E Z } ,
and B �f { en
e 27ria ,
: n E Z} ,
then: a) B is a subgroup of 2 in A; b) B n �B (/), and A = Bl:!�B; c) because a is irrational, both A and B are (countably) infinite dense subgroups of the compact group 'lI'. Zorn's Lemma implies there is a set 5 consisting of exactly one element of each 'lI'-coset of A. For M 5B , if MM- l n �B i- (/), i.e. , if
index
=
�
160
Chapter 4. More Measure Theory
since '][' is abelian, S 1 S;- 1 E �B C A. Hence, owing to the nature of 5, S 1 = 82 . Thus X 1 X;- 1 = b 1 b;- 1 E B, i.e., X 1 X;- 1 E �B n B = 0 , a contradic tion: MM� 1 n �B = 0 . If L is a measurable subset of M and T(L) > 0, then M M� 1 :J LL � 1 , which contains a ,][,-neighborhood of 1 (v. 2.3.9) and thus an element of the dense set �B, a contradiction. It follows that the inner measure of M, i.e., the supremum of the measures of all measurable subsets of M, is zero: T* (M ) = O. Because T is translation-invariant, T* (� M) = O. For x in '][' there is in 5 an s such that XS� 1 a E A. If x tJ. M, then a tJ. B: for some b in B, x = s �b E 5�B = �M. Thus '][' \ M c � M, and so T* ('][' \ M) = O. For each measurable set P ,
�
T * (P n M) + T* (P \ M)
(v. 4.5.3), whence T * (P n M)
=
T * (M)
•
T(P ) ,
T(P), in particular,
=
1 > 0 = T* (M) .
(}� 1 (M ) � M in JR. has properties analogous to those of M . The set M is nonmeasurable, ), * (M ) 0, and ), * ( M )
The set •
=
= 00 .
=
The set M is thick and for every measurable subset P of JR., ), * (P n M)
=
0 while ), * (P n M)
=
),(P)
(whence if )'(E) > 0, then M n E is nonmeasurable).
[
4.5.6 Note. The use of Zorn's Lemma or one of its equivalents is unavoidable in the proof that !fj(JR.) \ 5), i- 0 . Solovay [Sol] shows that adjoining the axiom: Every subset of JR. is Lebesgue measurable. to ZF, the Zermelo-Fraenkel system of axioms of set theory, yields a system of axioms no less consistent than ZF itself.] 4.5.7 Exercise. For (X, 5, f..l ) , if f..l* resp. f..l* are the associated �uter and inner measures induced by f..l on H(5), then a f..l * -a-finite E is in 5 iff
4.5.8 Exercise. a) If f : JR. r-+ JR. is (Lebesgue) measurable and p is a polynomial function, then p 0 f is measurable. b) If g E C(JR., JR.) , then g 0 f is measurable. c) For the Cantor set Co, the corresponding Cantor function
Section 4.5. Nonmeasurable Sets
161
'1jJo (v. cPo and the function 5
( �)
2.2.40 and 2 . 2 .41 ) are continuous and: c1) �f J contains a set that is Lebesgue nonmeasurable;
'1jJ (10 , 20 ) = 2 ' 1 1 l c2) '1jJ (5) �f g(5) is Lebesgue measurable; c3) h �f X ,p - l ( S ) is Lebesgue
measurable; c4) h o g is not Lebesgue measurable. In sum:
continuous function composed with a measurable func tion is measurable; a measurable function composed with a continuous function can fail to be measurable. [ 4.5.9 Note. For measure spaces (Xi , 5 i , f.-li ) , i = 1 , 2, the Fu A
binate (Xl x X2 , 5 1 52 , f.-l l f.-l2 ) derived in Fubini's Theorem is automatically complete (v. 2 .1.4 2 d ) ) and 5 1 x 5 2 :J 5 1 8 5 2 . There follow instances (4.5.10) where :J is = and others (4.5.18) where :J is �.l X
X
4.5.10 Example. If 5 i �f R or 5 i �f C, i = 1, 2, in 4.5.1, then
4.5.11 THEOREM. FOR FINITE MEASURE SPACES ( Xi , 5 i , f.-li ) , i = 1, 2, THE FUBlNATE (X l X2 , 5 1 5 2 , f.-l l x f.-l2 ) IN 4.4.9 IS THE COMPLETION OF THE PRE-FuBINATE (X l X2 , 5 1 8 5 2 , f.-l l 8 f.-l2 ) ' PROOF. In the context and notation of the general DLS construction of Chapter 2, L is li ll I -dense in L l ( X, f.-l). Hence, if E E D and f.-l (E) < 00, then X E is the II Il l -limit of a sequence {fn} nEN in L. In the current con text (cf. 4.4.2 and 4.4.5), each In is a simple function with respect to 5 1 8 5 2 . Via passage to a subsequence as needed, it may be assumed that lim In �f I exists a.e., and with respect to f.-l l x f.-l 2 , I � X E . If each In n---+ = is replaced by gn cl=ef 1 if In i- ' o otherwise X
X
X
{
0
lim= gn �f g exists a.e., and with respect to f.-l l then {gn } nEN C L, n---+
If Fn �f { (x, y) · l Fn n---+In = 1
=
clef F
x
f.-l2 ,
Chapter 4. More Measure Theory
162
exists and is in 5 1
x
5 2 , and f..l 1
x
(
f..l2 Ft:J.E
)
=
O.
D 4.5.12 Exercise. The conclusion to 4.5.11 remains valid if the measure spaces are a-finite. 4.5.13 COROLLARY. THE FUBINATE OF (JR.k , s T k , ),0k ) AND (JR.1 , S T1 , ),0 1 ) IS THE COMPLETION OF THE PRE-FuBINATE (JR.k+ l , S� (k+ l ) , ),0 ( k+ l ) ) . 4.5.14 Exercise. If L �f Coo (JR.n , JR.) and I is n-fold iterated Riemann in tegral: I : Coo (JR.n , JR.) 3 f r-+ f dXn . . . dX 2 dX 1 , then I is a DLS functional. The result of applying the DLS procedure in the con text of L and I is a measure space that is the n- fold Fubinate of (JR., 5)" ), ) with itself. 4.5.15 Note. The n-fold Fubinate of (JR., 5), , ),) with itself is denoted (JR.n , S),n ' ),n ) ; ),n is n-dimensional Lebesgue measure .
l (l C·· (l ) ) )
[
]
4.5.16 Example. For the set M of 4.5.5, 5
�f {O}
x
M C {O}
x
JR. E S� 2
and ), X 2 ({O} x JR.) = 0: 5 is a subset of a null set (),X 2) . If 5 E S�2, then = M E 5)" a contradiction: (JR.2 , ST2, ), 0 2) is not complete; (JR.2 , S�2 ' ), ) 2 is complete. 50
4.5. 17 THEOREM. IF {k, l} e N, THE COMPLETION OF
\
IS (TTll k+ l , 5 ), k + l ' A k+ l : S 0), ( k+l) 5 ),k + l ' PROOF. Since JR.n is totally a-finite with respect to both ),n and ),xn, the results in 2.5.13 and 2 .5.14 apply. D m.
)
-
4.5.18 COROLLARY. S� (k+ l ) � S� ( k+ l) . 4.5.19 Note. When n > 1 and confusion is unlikely, the notation (JR.n , 5),, )') replaces the unwieldy (JR.n , S)'n ' ),n ) and the associated usages are simplified.
[
]
4.5.20 Exercise. The measure space (JR.n , 5),, )') is translation-invariant and rotation-invariant, i.e., for T a translation or a rotation of JR.n , if E E 5)" then T E E 5), and )' T E = ), E .
( )
[ ( )] ( )
163
Section 4.5. Nonmeasurable Sets
[Hint: The a-ring generated either by the set of all open balls or the set of all open rectangles
n
contains 5 � .J
(lR.nn, x
4.5.21 THEOREM. IF 5), n ' f.-l ) IS A MEASURE SPACE SUCH THAT EACH HALF-OPEN INTERVAL k l [a k, bk) IS OF FINITE f.-l-MEASURE AND f.-l IS TRANSLATION-INVARIANT, =FOR SOME NONNEGATIVE CONSTANT p,
l) n )
[o, l) n
n
PROOF. If f.-l ( [0, is the union of 2 k pairwise �f p, then, since disjoint half-open subintervals Ij , each the translate of any other,
Thus f.-l ( E) = p for every half-open interval E and hence the equality ) obtains for every E in the a-algebra generated by the set of half-open ��. D As remarked earlier, 4.4.9 imposes no restriction on the underlying measure spaces. Since f E L l (X, f.-l ) , E,,,, (f, 0) is a-finite, i.e., the integration with respect to f.-l is performed over a a-finite set. However, if only 5measurability for f is assumed, K", (f, O) can fail to be a-finite and the conclusion of Fubini's Theorem can fail.
An ( E
4.5.22 Example. If X l = X2 = 5 1 = 5 )', 5 2 = f.-l l = ).., and f.-l2 is counting measure then E �f { (x, y) : x y } E 5 1 5 2 �f 5, the char acteristic function f �f X E of the diagonal E �f { (x, y) : x y } is 5measurable, and v,
!fj(JR.) ,
JR.,
=
X
=
In this instance, K", (f, O) is not a-finite. Indeed, f is not integrable with respect to f.-l l x f.-l2 .
164
Chapter 4. More Measure Theory
4 . 6 . Differentiation
The symbol
dy - used .
dx �
m
: S), 3 E
�: that appears in LRN is reminiscent of the classical symbol elementary calculus. If f E e ([0, 1 ] , q , y clef lX f(t) dt, and Ie f(t) dx � �(E), then � « >. and =
r-+
dy dx
lim >.([x ,x+h] ) -+O h>,o
>.([x,x+h])-+O h>,o
lim
=
d� d>'
0
J:+ h f(t) dt
(4 . 6 . 1)
h �([x, x + h] ) >.([x ' x + h]
(4.6.2)
- = f(x).
(4.6.3)
The display above suggests that for the measure space (JR., a complex measure space (JR., , p, ) such that p, « >.,
S),
>' ( E ) -+O >' ( E)>'O
lim
p,(E) .
S)" >.) and (4.6.4)
>.(E)
��
However, the left member of (4.6.4) is a point function, i.e., E ClR , while there is no reference to any point of JR. in the right member. The resolution of this difficulty is addressed in the following paragraphs. A particular con sequence of the development is a useful form of the Fundamental Theorem of Calculus (FTC) (v. 4.6.15, 4.6.16, and 4.6.33) : If f E L l (JR., >.) and F(x) �f
and F' � f.
lx f d>' , then F' exists
a.e. (>.)
In the wider context of Lebesgue measure and integration, FTC is a corollary to several more general results that will emerge as the discussion proceeds. [ 4.6.5 Note. Most of the next conclusions are true almost every where ( a.e.) , not necessarily everywhere. The calculations involve measures of unions of sets. Because measures are count ably addi tive, the arguments are eased if the constituents of the unions are pairwise disjoint. The context is a complex measure space (JR.n , p, ). The Jordan decomposition of p, and LRN permit the assumption that p, is nonnegative and totally finite, whence regular (v. 4.2. 19) . Thus attention can be focussed on open sets, since the measure of an
S)"
Section 4.6. Differentiation
165
arbitrary measurable set E can be approximated by the measures of open sets containing E. The following constructions allow much of the argument to be con fined to special collections of half-open cubes Q and associated open cubes C. . . . These are used well in a different context in Section 7.1 . ]
U
'"
as
�f (kl , . . . , kn ) in Zn , cl=ef { x : cl=ef (X l , . . . , xn ) , k; Xi k is the half-open cube vertexed at (k, m) cl=ef ( l m, >. [8 = 0, and thus >.(N) �f >. [ U For m in N and k
Qk,m
X
2m :::;
ki + 1 . , 1 :::; l :::; n r }, THEN U IS OPEN, ) i.e., M IS lsc. b) IF r > 0, THEN A(U) :::; If.-ll (JR.n . r PROOF. a) If r :::; 0, then U JR.n . If r > 0 and x E U, then for· some m, If.-ll [Ck= , m (X)] > rA [Ck= ,m (X)]. If Y E Ck= ,m (x), Ck= ,m (Y) = Ck= ,m (X) , whence M (y) > r, i.e., y E U: Ck= , m (x) C U. Since Ckm , m (x) is open, U is open . b) Each x in U is in some open cube C.. .(x) contained in U and such ] that I �I �:: (�� > r. If {Cp} EN is an enumeration of the cubes C... (x) P [ x varies over U, there is a subsequence {Cpq } qEN defined inductively as follows: WHEN
\ /\
=
as
167
Section 4.6. Differentiation .' CP1 = C 1 ; .' if CP1 , , •
•
•
Cpq have been defined, are pairwise disjoint, and u
=
:
u jq= 1 Cp
J
the induction stops. If U i- U = 1 Cpj , Pq + 1 is the least P greater than q Pq and such that C is disjoint from U = C J 1
.
p
p
J
. •
Since two cubes C . are disjoint or nestle, U = U
Cpq and for each q, I IL I/Cpq/ > r. Thus )'(U) "' ), (Cpq ) '" IILI (CpJ ::::: IILI (l�n ) . D �q r �q r ), Cpq 4.6.12 Exercise. If f E L 1 (l�n , IL), IL : 5 3 E r-+ IL(E) �f Ie f d )" and =
r } ) ::::: ll1.lhr . 1 :$m m---+ = ).. [Ck= , m (X)] o C o ).. ) > n Coo (lR, , q a g such that I l f - g il l �f I l h l l < E. Furthermore, ).. [Ck:,m (X)] ik=. = (X) I f(Y) - f(x) 1 d)" (y ) ' = (X) I h(y ) 1 d)" (y ) + Ih(x)1 � ).. [Ck:, m (X)] ik= + ).. [Ck:,m (X)] ik= . = (X) I g (y ) - g(x)1 d)" (y ) clef Im + IIm + IIIm' r
r
=
However, -
r
m---+ = IIm > -3 } lim
Owing to 4.6.13, the last of the three summands above is empty; owing to 4.6.11 and 4.6. 14, the measure of each of the first two summands does 3E not exceed - . D r
4.6.17 Exercise. a) For some x-free constant Kn , if x E lR,n \ N and there is a
Ck,m (X) containing B(x,
r
r
> 0,
t and such that
Section 4.6. Differentiation
169
b) For some x-free constant L n ,and each Ck, m (X), there is a B(x, rt con taining Ck , m (x) and such that
> L n >. [Ck1,m (X)] '
1
>. [B ( x, r)O]
4.6.18 Exercise. If f E L l (lR.n , >.) and r ..(. 0, for all x off a null set,
1 1 I f(y) - f(x) 1 d>.(y) = 0, f(x) = m . >. [B(x,1 r)0] 1 f(y) d>.(y).
hr+O p
.
>. [B( x, r )0]
B (x,r) O
hrto
B (x,r) O
[Hint: For Ck ,m (X) and B(x, rt as in 4.6.17a)
K" >. (B(x,1 r) ) }r 1 (x)] 1 >. [Ck ,m 0
B (x , r) o
�
If( y ) - f(x) 1 d>.( y )
Ck,m (X)
If( y ) - f(x) 1 d>.( y ).]
[ 4.6.19 Remark. In [Rud] nicely shrinking sequences and in [Sak] sequences with parameters of regularity are introduced to define alternative versions of Df.-l. Since two sets A and B under lying those treatments can fail to be disjoint or to nestle, covering theorems related to Vitali 's Covering Theorem are used to cope with this situation. The cubes C... require no appeal to such de VIces. The burden of 4.6.17b) is that for each x in not in N there is a nicely shrinking sequence {Ck ,m (X)} m EN ' Similarly, 4.6.17a) says that for each x not in N there is a sequence { B (x, rm ) ° } mEN that is nicely shrinking relative to some sequence in Q.]
There remains a collection of results dealing with three classes of func tions specified in the terms and symbols of 4.6.20 DEFINITION. WHEN P IS A RIEMANN PARTITION (cf. 2.2.50) OF [a, b] AND h E ClR : n a) THE P-variation OF h IS varh (P) �f L I h -h THE total
variation OF h ON [a, b] IS
k=2 (Xk) (Xk- dl ;
P a Riemann partition of [a, b] };
Chapter 4. More Measure Theory
170
b) WHEN var h([a, b]) < 00 , h IS OF bounded variation on [a, b] :
h E BV([a, b]) ; WHEN
sup
-oo< a < b x E D and F (x') > lim F(y) y=x
x' tJ- N(x), and for some N1 (x) , sup F(y)
clef =
Lx : F (x') > Lbn • But (4.6.27) implies x' E (x, bn l . If c �f sup { x' : x < x' :s; bn , F (x') > L x }, then an < x < c :s; bn . If c < bn , then for some c', c' > c, and As in the previous argument, c' E (c, bn l. However, by definition of sup, c' :s; c, a contradiction. Hence c = bn and L bn 2: F (bn ) 2: Lx > L bn , a final contradiction. Thus F( x) :s; L bn ' D
Figure 4.6.1.
x-axis
Section 4.6. Differentiation
173
[ 4.6.28 Remark. Riesz's result is applied below to the proof that monotone functions are differentiable a.e. Its analog for se quences leads to a perspicuous proof of G. D. Birkhoff's Pointwise Ergodic Theorem, [Ge3] .]
If f E ffi.lR and x i- y, the ratIO gy (x) question of the existence of f'. •
clef f(y) - f (x ) is central to the =
y-x
4.6.29 THEOREM. I F f E MaN, THEN f ' EXISTS OFF A NULL SET. PROOF. If
�f limg y (x), Dl ( X ) �f limgy (x), ytx ytx DT(x) �f limgy (x) , DT (x) �f limgy (x), ytx ytx D1 (x)
then 0 :::; Dl (X) :::; D1 (x) :::; 00 and 0 :::; DT (X) :::; DT(x) :::; 00. If a) DT(x) < 00 a.e. and b ) DT(x) :::; Dl ( X ) a.e., then a ) and b ) apply to x H f( -x), which is also in MON. Hence 0 :::; D T (x) = Dl (X) < 00 a.e. and the argument is finished once a ) and b ) are proved. a ) . If Eoo { x : DT(x) = oo }, then for each x in E, any positive A, and some y such that y > x, gy (x) > A. Hence, for the map -
�f
F : x H f (x) - A x and its associated set D in 4.6.26 , there is a sequence { (an , bn )}l :s; n < M :S; oo depending on A and such that D = U < n<M < oo (an , bn ) . Furthermore, lE c D and -
F ( an ) :::; lim F ( x ) , A (bn - an ) :::; lim f ( x) - f ( an ) , x =bn
x=bn lim f ( x ) - f (an ) :::; f( l ) - f(O) . (bn - an ) :::; A l :S;n<M:S;oo l :S;n<M:S;oo x =bn
L
L
' /\' (D) < f( l ) - f(O) , and SInce ' cont alne S·Ince E IS ' ' d In ' the open set D , sInce A A may be arbitrarily large, >.(E) = O. b ) . If _
E
.).
f(l) - f(O) :
lx
b) Both 9 [0, 1] '3 x H f(x) - f(O) and h : [0, 1] '3 x H f'(t) dt are in MON([O, 1] ) . c ) According to 4.6.18, g' � f' � h'. d) If fs = 0, then f(x) = f(O) + f'(t) dt, whence f E AC([O, 1]). If f E AC([O, 1]), then 4.6.15 and 4.6.18 imply fs = o. e) The argument based in a) on Fatou's Lemma may be repeated to f'(t) dt. Furthermore fs 0 iff f E AC ([O, 1] ) . imply f(x) - f(O) 2': D 4.6.34 Example. a ) If :
lx
lx
f(x) �f
=
{�n(x)
if O < x � 1 if x = 0
then f'(x) � .! and f' tJ- L 1 ( [O, 1] , >.). b) If f = x
x E [0, 1] ,
lx f'(t) dt
=
cPo , then f' � 0, whence if
0 and f(x) = fs (x) .
4.6.35 Exercise. If - 00 < a � b < 00 , assertions like those in 4.6.33 are valid for functions in BV ( [ a, b] ). 4.6.36 Example. If f(x) == x, then f E MON n AC and f' == 1. Hence f' tJ- L 1 (JR., >' ) , and the formula f(x) = fs (x) + f'(X) dx is invalid.
[Xoo
4.6.37 Exercise. If f' exists everywhere on [ a, b] and f' (a ) < C < f' (b) , then for some � in (a, b) , f'(�) = C. A derivative enjoys the intermediate value property.
Chapter 4. More Measure Theory
176
[Hint If g(x) �f f(x) - ex, then g'(a ) < 0 and g' (b) > O. Hence, for some � in ( a , b), g(�) = min g(x ) .] :
as;xS;b
4.6.38 THEOREM. IF: a) f E ]R.[O, 1 l ; b) I' EXISTS THROUGHOUT [0, 1] ; AND c) I' E L 1 ([0, 1] , ),) , FOR EACH x IN [0, 1 ] , f(x) -
f( O ) =
1x f'(t) dt.
[ 4.6.39 Note. The result is in sharp contrast with 4.6.34e) . Thus the assumption b) provides the crucial ingredient for the validity of the conclusion.]
PROOF. The line of argument is to show that f(l) - f( O ) �
11 I' (t) dt. 1x t(t) dt.
That conclusion is applicable to - f and leads to f(l) - f( O ) = The same kind of reasoning can be used when 1 is replaced by any x in [0, 1] .
Although, as a derivative, I' enjoys the intermediate value property, that alone does not suffice for the present purposes. In [ GeO] there is an example of a null function that assumes every real value in every subinterval. The technique is rather to exploit the lower semicontinuity of an aux iliary lsc function 9 that approximates I' from above. The DLS construction implies that if E > 0, there is in lsc([O, 1] ) a 9 such that 9 2': I' and
11 °
g dt
lx+8 p(t) dt - f(x J) f(x) EJ + + + x
J'(x) J - J'(x) + E J + EJ = 0,
[
]
in denial of (4.6.40). 4.6.41 Remark. Many more theorems about derivatives can be found in the literature, e.g., (Denjoy-Young-Saks) If f E for some set A, )'(A) = 0, and if x tJ- A, then
[
D
ffi.lR ,
•
f ( x +_ )h'--- f (x + h) - f (x) = = lim _ _f-'...(x--'-) lim ' h h "tO h "tO h or f' (X) exists, 00
-,---_
_
v. [SzN] ; In the metric space ( O , the set of nowhere differen tiable functions is of the second category, cf. 3.62 in [Ge3] .]
C [ 1], ffi.) ,
•
4.6.42 Exercise. If f E BV and for some g , f = g' , then f E
C(ffi., C).
4 . 7 . Derivatives
The developments in Section 4.6 lead to a rigorous proof of the change of variables formula of multidimensional calculus. . cle For x cle =f (X I , . . . , Xn ) and y =f ( Y I , . . . , Yn ) In ( x, y ) cle=f '" � XkYk mm m. ,
n
n
k= 1
and I lxll � �f L x� .
k= 1
ffi.n m'3 x n [ffi. , ffi. ],
4.7.1 DEFINITION. A FUNCTION f entiable at a IFF FOR SOME Ta IN
:
H
Y �f f(x) E
f(a + h ) - f(a) - Ta ( h ) 11 2 lim Il II h l 1 2
h -+ O h >' O
=
0;
ffi.m IS differ
eo
Chapter 4. More Measure Th ry
178
I AT a: T.a 1 - E > I l y 1 2 , i.e., not only is y not in (B(O, I n but y is not in g[B(O, 1)] . Thus G : x II yy - g/x)X)II 2 IS a well-defined continuous map of B(O, 1) into itself. Brouwer's Fixed Point Theorem (1.4.27 and 1.4.36) implies that for some xo , G (xo) = Xo . If IIxol1 2 = 1 , the equation G (xo) = Xo and Schwarz's inequality [(3.2. 11)] imply I l y - (xo) 1 2 = (xo, Y - (xo)) = (xo, y ) + (xo , x - (xo)) - 1 < Ilyll + E - 1 < 0, a contradiction. If I xol1 2 < 1, since II G (x) 11 2 == 1 , Xo i- G (xo), a final con tradiction. D a') (T is nonsingular) By definition, if E > 0, for a positive J, '*
9
9
H
-
9
9
9
9
'
As a consequence,
1 /(x)1 1 2 < (1 + E) l xI 1 2 , i.e., 1 [B(O, rt] C B[O, (1 + E)r] o . By virtue of 4.7.10, if E E (0, 1), then B(O, ( 1 - E)rt C 1 (B(O, rn. A { [B(x, r )O] } Hence, if E E (0, 1) , then (1 - E) n � J (1 Et and (4.7.9) A [B(x, r) O ] � + obtains. b') (T is singular) If E > ° and U, �f { x : inf I l x - Y l 1 2 < E } ' yElm (T) then U, is the open set consisting of all x within a distance of E from
some point in im (T) . If K is compact, K n U, is covered by finitely many
Chapter 4. More Measure Theory
180
CK , A (K :::; C ro , r :::; ro CK, :::; CK2 . Ar I l f(x) - J'(O)x I 1 2 :::; Ar l x l 1 2 , I x l 1 2 < r, Kr rtO Ar [B(O,r)O]} -< ArCKrn rn ArCKr . f[B(O , r)] , A {AJ[B(O, D r r)o] [ 4.7.12 Note. In (4.7.9) each B(x, rt may be replaced by an open cube Cr(x) with edges parallel to the coordinate axes, edge length 2r , and containing x. Corresponding to (4.7.9) is A {g [Cr(x)]} [f'(x)] . r� A [Cr(x)] There is a constant Kn such that for all x in ffi.n , the norm open balls of radius E. Hence, for a constant KEn . n U,) C Furthermore, if then For a positive if and there is an such that and lim = O. The facts just stated imply that when is the compact set
Kl K2 ,
=
1·
=
P
I l x l ' :::; IIxl 1 2 :::; Knl l x l ' , i.e., I I ' and I 1 2 are equivalent . x and of radius r for the norm I I ' is B'(x, rt �f { y : I l x - yll' < r } ,
satisfies The open ball centered at
i.e., the open cube with edges parallel to the coordinate axes, edge-length and centered at Corresponding to 4.7.10 is the statement:
2r,
If 9 E then 9
x.
C (B'(O, 1t,ffi.n ), E E (0, 1), and {l l x l ' = I} {llg(x) - xi i ' < E} , (B' (0, 1 n B' (0, 1 Et .] ::)
'*
-
4.7.13 Exercise. The assertions in 4.7.12 are valid. Together with 4.7.6, 4.7.14 below forms the basis for the change of variables formula of multidimensional calculus (4.7.23 below).
N IS A NULL SET (A) IN ffi.n , f E (ffi.n ) N , AND FOR ) f ) IN N , inf sup I l f (� - fr 11 2 < 00 , THEN f(N) IS A NULL N(y) N(y)3x#-y Ix- 2
4.7.14 LEMMA. IF EACH SET
Y
(A).
Y
Section 4.7. Derivatives
181
N is the union of the count ably many sets Ekp �f { X : x E N, I l f (x) - f( y ) 11 2 � kllx - yI1 2 , E B (x, � ) n N } , k ,p E N, if each f (Ekp ) is shown to be a null set ( ), ) , the result follows. For the rest of the argument, the subscript kp is dropped. If E, 1] > 0, E is contained in an open set U such that ), (U) < 1]. As in the PROOF of 4.6.8, U is the union of pairwise disjoint half-open cubes Q ... and L... ), (Q .. . ) < 1]. Since each half-open cube Q ... is the union of PROOF. Since
Y
all the smaller half-open cubes it contains, E ( = Ekp ) can be covered by pairwise disjoint half-open cubes of diameter less than � . If ... E E , then ... there ob ... , diam . . . ) t , whence for some constant tains for the sum of the measures of all the open balls [x.. . , diam ... W corresponding to all the ... in all the Q ...
Q C B [x
p x n Q ... K, (Q B
(Q Q
L), {B [x... , diam (Q . . . ) t } K L ), (Q ... ) < K1]. =
Hence, if
1] �, then L ), {B [x. . . , diam (Q . . . ) t } < E . =
According to the definition of E ( = Ekp ) ,
f(E) C U B [f (x... ) , k · diam (Q . )] , whence
L ), {(B [f (x... ) , k · diam (Q ... )]} � kn L ), { B [x.. . , diam (Q . . . ) t } D
V f' V, f () () f(V): {{,>, (N) o} 1\ {N C V}} {,>, [f(V)] o} ;
4.7.15 COROLLARY. IF IS AN OPEN SUBSET OF ffi.n AND EXISTS AT EACH POINT OF THEN: a) MAPS NULL SETS ), CONTAINED IN INTO NULL SETS ), CONTAINED IN
f
=
'*
V
=
b) MAPS LEBESGUE MEASURABLE SETS INTO LEBESGUE MEASURABLE '* {J(E) E SETS: {E E PROOF. a) 4.7.14 applies.
S ), }
S ), }.
Chapter 4. More Measure Th ry eo
182
b) Since ), is regular and JR.n is a countable union of compact subsets n (JR. is a-compact) , if E E S)" then for some a-compact set 5 and a null set N, E 5'0N. Hence f(E) f(5) U f(N). Moreover, f(5) is a-compact; owing to a), [f ( N ) ] O. D 4.7.16 E xercise. There is no conflict between 4.7.15 and the following facts: a) the Cantor function cPo is differentiable a.e. ( ),); b) the Cantor set Co is a null set ( ),); c) ), [cPo (Co )] l. = 4.7.17 Exercise. a) If f E (JR.n ) lR f (x) �f (/1 (x), . . . , fn (x)) and J' ex8-fi (x) eXIsts, . . then . 1 < l. < n , 1 < . < m, and t he matnx ' representmg Ists, 8Xj the linear transformation J' with respect to the standard basis e l , . . . , en is ( 8f8xj(X) ) j �f J(I) , J' (x ) I S i S n, I S Sm the Jacobian matrix of f. b) The entries in the Jacobian matrix J( I ) are = Lebesgue measurable functions. c) If f E [lRm , JR.n ] (c (JR.n ) lR ) and n f (ei ) �f jL aijej , 1 � i � m, "" l 8fi (X) 1 � l. � n, 1 � . � m . then aij ---' 8X · 4.7.18 THEOREM. IF: a) f E (JR.n ) B ( O , 1) o; b) J' EXISTS THROUGHOUT B(O, It; c) f IS INJECTIVE AND sup { llf(x)1 1 2 : x E B(O, It } < 00; d) g : JR."' f--t [0, (0) IS LEBESGUE MEASURABLE; e) p IS THE FUNCTION IN 4.7 .6, THEN BOTH (I' ) 1 B ( O, I) O AND g o f . p ( I' ) 1 B ( O, I) O ARE LEBESGUE MEASURABLE AND J f ( B (O, I) O ) d)' irB ( O, 1) 0 (g f) . p ( I' ) d)'. PROOF. The measurability of p ( l' ) I B ( o , 1) 0 is a consequence of the results in 4.7.17 . If E E S)" 4.7.15 implies: a) p,(E) �f ),[f(E)] exists; b) (JR.n , S)" p,) is a complex measure space (direct calculation); c) p, « ),. Owing to 4.6.12, if E E S ), n B(O, It, then p, (E) Ie Dp, d)'. Hence, for x in B(O, It and small r, p, [B(x, r t] ), [f (B(O, r)O)] (4.7.19) ), [B(x, r)o] ), (B(x, r)O ) Owing t o 4.7.6 and (4.7.19), Dp,(x) � p [J' (x) ] (v. 4.7.12). =
),
=
=
=
_
=
=
_
_
J
_
i
J
J
P
=
9
=
0
Section 4.7. Derivatives
183
If A E S,B and E �f f 1 ( A ) , then X E = X A 0 f: E E S,B ( C SA ) . Thus -
J
X
f(B(0, 1) 0)
A
d)" = r X 0 f . p (I ' ) d)" . lB(0, 1)0 A
( 4.7.20)
If N is a null set ().. ) , for some A in S,B, A ::) N and )"(A) o. Thus (4.7.20) is valid with A replaced by any Lebesgue measurable subset E of B(O , It. The standard approximative methods extend the validity of (4.7.20) to the case where X A is replaced by an arbitrary nonnegative Lebesgue measurable g. D [ 4.7.21 Note. The PROOF of 4.7.18 shows that g o f . p (I' ) is measurable; g o f need not be measurable (cf. 4.5.8).] =
4.7.22 Exercise. If: a) V is open in ffi.n ; b) X is a Lebesgue measurable subset of V; c) f V '3 x ffi.n is continuous, f is injective throughout X, and f ' exists throughout X; d) )"(V \ X) = 0; and e) 9 ffi.n [0, (0) is Lebesgue measurable, then
f--t
:
J
f (V)
9
:
d)" = r l
v
90
f--t
f . p ( I' ) d)".
[Hint: The open set V is the countable union of open balls; 4.7.18 applies.]
4.7.23 THEOREM. IF f E [ffi.n ] AND f ' EXISTS, THEN p [!, (x)]
=
Idet {J[f(x)]} I .
PROOF. If T �f !' (x), e is a constant, and
if k = i otherwise '
then I det { J [f (x)]} I = l ei . On the other hand, if the edge-lengths of a cube C .. are S 1 , , S n and the corresponding edge-lengths of T ( C. ) are s�, . . . , s � , then sj = sj if j i- i, and s ; = l ei · Si, whence p(T) = l e i : Idet {J[f(x)]} 1 = p(T) . If if k sl {i, j} if k = i if k = j ..
.
•
•
•
Chapter 4. More Measure Theory
184 T(ABCD) A(ABCD)
=
=
ACED
A.(ACED), - "
E
, ,
,
D t-C---'--+----.,.,, C ,
A
Figure 4.7. 1.
B
then l[f(x)] is the result of interchanging rows i and j of the identity matrix: Idet {l[f(x)]} 1 = 1. Because the measure of any cube C... of edge length £ is regardless of the labeling of the coordinate axes, p(T) = 1: Idet {J [ f (x) ] } I = p(T) . If T (ek ) = ei + ej if k = i otherwise ' ek
£n ,
{
then Idet {J [f (x)] } I 1. If n = 2, by direct calculation (integration or elementary geometry), (C. . . ) = [T (C . . . )]. The situation is illustrated in Figure 4.7.l. If n > 2, via Fubini's Theorem and induction, (C. .. ) = [T (C... )]: Idet {J [f (x)]} I = 1 = p(T) . Since every linear transformation is the composition of a finite number of transformations, each like one of the three just described, the product rule for determinants applies [Ge2] . D 4.7.24 Exercise. a) If + > 0, the map =
),2
), 2
),n
),n
a 2 b2
is bijective and !, exists. b) What is p ( I' ) ? When (X, S, p, ) and (Y, T, � ) are measure spaces an F in yX is defined to be measurable iff { E E T} ::::} {J - l (E) E S } [Halm] . In the current context, since [ffi. ] may be regarded as , !' as a function of x may be ]Rn regarded as an element of
n
(ffi.n2 )
ffi.n2
4.7.25 Exercise. Both f ' and p ( I ' ) are Lebesque measurable.
Section 4.8. Curves
185
4 . 8 . Curves
4.8.1 DEFINITION. A curve IN A TOPOLOGICAL SPACE X IS AN ELEMENT "/ OF C([O, 1] , X ) . THE SET im h) �f "/* �f { "/ ( t ) : t E [0, 1] } IS THE image of"/. WHEN X IS A METRIC SPACE (X, d) , THE ( POSSIBLY INFINITE) length fih) OF "/ IS
=
WHEN fih) IS FINITE, "/ IS rectifiable. WHEN ,,/ ( 0) ,,/ (1 ) , "/ IS closed. WHEN "/ �f hI , . . . , "/n ) E C ( [O, 1] , ffi.n ) , f E C h* , c) , AND "/ IS RECTIFI ABLE, THE Riemann-Stieltjes integral of f with respect to "/ IS
4.8.2 Example. For the curves
"/ 1 : [0, 1] 3 t
f--t e 27rit E C, "/2 : [0, 1] 3 t f--t e 37rit , "/3
"/ � = "/� = ,,/; , while fi h d is not a closed curve.
=
271", fi (2 )
=
371", fi (3)
=
:
[0, 1] 3 t
f--t e47rit ,
4 71" . Furthermore, "/2 =
+
4.8.3 Exercise. For the Cantor function 0, then for some P, £ ("!p » £("!) - E; c) {£ (,,/p » £ ("! ) - E} ::::} {l hp - "/ l oo < E}. d) if n = 2, f E C ( [0, 1], C) , and E > 0, for some P, I I I f d,,/p - 1 1 f d"/ I < E. [ 4.8.6 Note. The curve ,,/p is piecewise differentiable; on each interval (tk , t k+ d, "/� is constant. The set b (t k )} l -< k -< n of its vertices is a subset of ,,/*. If f E C([O, 1], C) , then
f d,,/ is eased if "/ is well 1 behaved or, like ,,/P , only piecewise well-behaved. For example, if
More generally, the calculation of
{tk } l :$k :$n is a Riemann partition of [0, 1] and
then
1 f dz = 11 f
0
"/ . "/' dt .]
,,/ �f a + ib, {a, b} c C ([O , 1], ffi.) , THEN "/ IS RECTIFI {a b} BV([O, 1]).
4.8.7 THEOREM. IF ABLE IFF , C
PROOF. The triangle inequality (v. 3.2.12) implies that
{ I a (tk+ d - a (tk ) 1 , I b (tk+ d - b (tk ) l } ::; b (tk+ d - "/ (tk )1 ::; l a (tk+ d - a (tk )1 + I b (tk+ d - b (tk )l · Thus "/ is rectifiable iff {a , b} c BV ([0 , 1]). [ 4.8.8 Note. If "/ is rectifiable, then £ ("! ) is not only n- l sup L h (t k+ d - "/ (tk ) 1 0 = to < t l < ' " < tn 1, n E N , k=O max
{
=
:
}
i.e., the supremum of the lengths of the associated polygons with vertices on
"/* but, by abuse of notation, £ ("! )
=
11 1 · I d"/ I .]
D
Section 4.9. Appendix: Haar Measure
187
4 . 9 . Appendix: Haar Measure
A
topological group G is a Hausdorff space and a group such that G G '3 (x, y) xy - l E G X
H
is continuous. For a locally compact group G, there is a measure defined on aR[K(G)] �f S(G) . For there obtains:
- Haar
measure-p, p, {{ E E S(G)} !\ {x E G}} {{ xE E G} !\ {p,(xE) p,(E)}} , i.e., p, is translation-invariant. Complete proofs of the existence of Haar measure p, and derivations =
'*
of its most important properties are given in [Halm, Loo, Nai, We2] . Below is an outline of the fundamental idea behind the existence proof.
a) The motivation lies in the next alternative definition of the Riemann and Nf E N there is a nonempty family C integral. If f E Coo of nonnegative constants such that of sets
(ffi., ffi.+ ) { cn} - Nt ::;n ::;Nt (2Nt ) 2_ 1 f(x) � L k=O
Furthermore,
1 f(x) dx lR
=
Nt i�f L cn · n= - Nt
The integral stems from the
majorization of f by linear combinations of translates of x . b) Haar's idea was to imitate the procedure described in a), namely to majorize one function by linear combinations of translates of another and thereby approximate some kind of DLS functional. For a locally compact group G and a pair f, in Coo (G, there is a (possibly empty) family C of sets of nonnegative of G constants such that for some subset S �f
ffi.+ ) { g} { cn } - Nt - 0, AND
(I, g ) E U (G, /-l ) *
X
r
L q (G, /-l ) ,
THEN Il f gil T :::; Il f ll p . Ilgll q · 1 1 [ 4.9.8 Remark. If = then - + 1 and, by abuse of no tation, = 00. Then the conclusion of 4.9.7 reduces to (4.9.4).]
P q=
q p' ,
r
-
=
. 1ent to t he · -1 + 1 1 clef -1 > 0 IS. 1ogica11y eqmva P ROOF. The cond ·Itlon assumption that there are positive numbers (3, such that
P q -
1
1 (3
"I = I, P 1
-+-+o
1
-
1
r
0,
1 1 (3 '
q
0
1
0
"I
1
"I
,
0
Thus the factorization
=
r.
(4.9.9)
I f (y - l X) 1 ' lg (y) 1 = I f (y - l X) I ;; Ig (Y ) I � ' I f (y- l x) I P ( � - ± ) . lg (y) l q ( i - ± )
(
)
(three factors ) and 3.7.16 when *
I f g (x) 1 :::;
[L
0
(
P
= I , (3 =
)
P2 , "I = P3 imply
I f (y - l X) I P .* Ig (y) l q d/-l (y)
. [L
(
]
)
1
;;
I f (y - l x) I Pi3 ( � � ± ) d/-l(y)
]
1
i3
(4.9.10)
Of the last two factors in the right member of (4.9. 10) , the former is inde pendent of x ( because /-l is translation-invariant ) and is I f li p ; the latter is Ilgll q . The translation invariance of /-l implies also that
Direct calculations using the relations (4.9. 10) lead to the result. [ 4.9. 11 Note. The derivation above is based on nothing but careful use of Holder's inequality and its extension. In 1 1.2.13 the same result is seen to be one of several consequences of the M. Riesz Convexity Theorem, 11.2.6, of fundamental importance
D
192
Chapter 4. More Measure Theory
in the study of Fourier series and Fourier transforms, the basic ingredients of harmonic analysis on locally compact abelian groups [Loo, We2] .]
�f
{O})
4.9.12 Exercise. a) The set C* (C \ is a locally compact abelian group when its topology is that inherited from the customary topology of C and multiplication (of complex numbers) is the group binary operation. 1 b) The map f.-l : S), (C* ) 3 E H Jr 2 dx dy is a Haar measure for C* . E x + y2 [Hint: The discussion in 4.7.24 applies.] 4 .10. Miscellaneous Exercises )'
n
4.10.1 Exercise. (The Metric Density Lemma) If E E S ' then 1. ). [E n B (x, r )] 1m rtO ). [B(x, r )] ----'-:--;-:=-:--'-..,.,--'-'-
exists a.e. The limit is 1 for almost every x in E and is ° for almost every x in ffi.n \ E. [Hint: The results 4.6.11, 4.6. 14-4.6.17 apply when f X E .] 4.10.2 Exercise. If n = 1, 4.7.18 can be proved without recourse to Brouwer ' s Fixed Point Theorem.
�f
4.10.3 Exercise. If
{an } nEN C ffi., {dn} nEN C (0, 00), and if t < a if t 2: a '
00
�f L dnjan is in MON and Discont ( I ) {an } nEN ' n= l For in (0, 00) and a Lipschitz function (v. 3.2.31)
then f
=
a
L : ffi.2
3
{x, y}
H
L(x, y ) E (0, 00),
an f in ffi.lR is in Lip (L, O' ) iff If(x) - f(y) 1 :::; L(x, y) l x - y in . 4.10.4 Exercise. a) If L is a constant and f E Lip (L, 1 ), then f E AC. The converse is false. b) If f E C l ([O, 1] , ffi.) , for some constant Lipschitz function L, f E Lip (L, 1) . The converse is false. 4.10.5 Exercise. If {J, g } c AC, then { J ± g , fg } C AC. For some f in AC and some g in Coo (ffi., C) , fg tJ- AC.
Section 4. 10. Miscellaneous Exercises
193
4.10.6 Exercise. a) If 1 E AC ([a, b] ) and 9 E C([a, b] , q , then
ib
b) If
9
dl
=
{
ib
9
.
f' d).. .
1 (x) �f O �f -00 < x :::; 0 , 1 1f O < x < 00 1, then 1 E BV and 9 E C(JR., q , and g ( O) =
1=
l
9
dl >
l
9
. f' d)..
=
O.
c) For the (continuous) Cantor function cPo :
4.10.7 Exercise. If 9 E C([a, b] , q and h E BV([a, b] ) , then: a) exists; b) the following formula for integration by parts is valid:
ib g dh
[Hint: a) Only the situation 9 E C([a, b] , JR.) , h E MON([a, b] ) need be addressed. b) The formula (Abel summation) n- 1
L1 � ) [ X ) k= 1 ( k g ( k+I =
-9
Xk
( )]
) - 9 (x I ) 1 (6 ) 9 (x n ) 1 (� n -
I
n- 1
L g X ) [I �k ) - 1 (�k- I ) ] k=2 ( l,; (
applies to the Riemann sums approximating 4.10.8 Exercise. If 1 E C([a, b] , JR.) ,
F(t) then: a) F E BV([a, b] ) ; b)
=
9
ib 1 dg . ]
E BV([a, b] ) , and
it 1 dg , t E [a, b] , =
F(t) - F(t-) I(t) [g (t) - g (t-)], F( H ) - F (t ) = l (t) [g (H) - g (t)];
Chapter 4 . More Measure Theory
194
c) F' � f · g' .
4.10.9 Exercise. If U E 0 (JR.n ) , for a sequence { Kn} nEN of compact subsets: a) Kn = U; b) Kn C K�+l c) if K (JR.n ) 3 K c U, for some N, nEN
U
Kc
N
U Kn .
n= l
[Hint: For each n and m in N, the union of the finitely many Rk,m (v. Section 4.6) contained in B(O, nt n U is a compact subset of U. If K is a compact subset of U, and d is the Euclidean metric for JR.n , then inf { d (x, u) x E K, u E U } > 0.] :
4.10.10 Exercise. If then flh) = 11"1' 11 2 dt.
11
"Ii E AC ([O, 1] ) , 1 � i � n, and "I
clef =
( "1 1 , . . .
, "In ) ,
4.10.11 Exercise. If G is a topological group and H is an open subgroup of G, then H is closed (whence, if H ¥- G, G is not connected) . [Hint: Each coset of H i s open.]
4.10.12 Exercise. The value of a M in (4 . 9.2) is independent of the choice
of f.
4.10.13 Exercise. If G is a locally compact group with Haar measure p, and for each neighborhood V of the identity of G: a) Uv is a nonnegative uv (x) dp,(x) = 1 , the net function in A(G); b) Uv = ° off V; and c)
i
V H Uv f converges [in A(G)] to f and ( uv ) a ( a ) as a function of a , converges uniformly to ( a , a ) . [ 4.10.14 Note. The Banach spaces L l ([O, 1], >.) and L l ([O, 1), >.) are essentially indistinguishable since >. ({I}) 0. The map *
=
is a continuous bijection between [0, 1) and 'lI'. The topology T f { E : E C 'lI', \11 - 1 (E) E O{ [0, I)} } is that inherited by 'lI' from JR.2 and with respect to T, 'lI' is a topological group. The measure spaces ([0, 1] ' 5,6([0, 1]), >.) and ('lI', 5,6 ('lI') , T) are isomor phic via the bijection \II . Thus L l ([O, 1] , >') and L l ('lI', T) are iso morphic in the category of Banach spaces and continuous homo morphisms.]
�
195
Section 4. 10. Miscellaneous Exercises
4.10.15 Exercise. With respect to convolution as multiplication, i.e., with respect to the binary operation
the Banach space L 1 (1I', T ) is a Banach algebra A(1I') . 4.10.16 Exercise. If G is a locally compact group, f..l is Haar measure, 2: 1, E > 0, and U(G, f..l ) , in N(e) there is a V such that
IE
p
is the set of Fejer's kernels (
4.10.17 Exercise. If {FN}
v.
3.7.6) and
I E U([O, 1 ] , ), ) , then FN INEN exists for each N and I I O. Nlim -t oo II FN - l p [Hint: The result is true if I E C( [O , 1] , q .] n 4.10.18 Exercise. If {J, g } c (ffi,m ) lR and both I ' and g' exist at some n a in ffi,n , then: a) h �f (f, g) E ffi,lR ; b) h ' exists at a; c) *
h ' (a)
=
*
(f, g) ' (a)
=
=
(f' (a), g) + (f, g' (a) ) .
[Hint: (f(a + x) , g(a + x) ) - (f(a) , g(a) ) ( f (a + x) , g(a + x) ) - (f(a + x), g(a)) + (fa + x) , g(a) ) - (f(a) , g(a)) .] =
4.10.19 Exercise. If E c ffi, and )'(E) = 0, then ffi, \ E is dense in ffi.. 4.10.20 Exercise. For 1 m2 '3 (X1 , X2 : m.
) {
how do the iterated integrals
H
(x i - x� ) ( x i + x� ) O
if X21 + x 22 > 0 otherwise
Chapter 4. More Measure Theory
196
compare? 4.10.21 Exercise. For the measure space (X, 5, f..l ) that is the Fubinate of ( X l , 5 1 , f..l d and ( X2 5 2 , f..l2 ) , 5 contains the a-ring 5 consisting of all empty, gnite, or countable unions of sets of the form E l X E2 , Ei E 5 i , i = 1 , 2. Is 5 necessarily 5? [Hint: The case (Xi , 5 i , f..li ) = ( [0, 1] , SA, >') , i = 1 , 2, is relevant.] '
4.10.22 Exercise. If
X I - X 2 E Q } and {A l , A 2 } C (5A2 n E) ,
E = ffi? \ { (X l , X 2 )
then >. ( A I X A 2 ) = 0. 4.10.23 Exercise. Is there a signed measure space ([0, 1] , 5, f..l ) such that f..l =t=- 0, f..l « >., and for all a in [0, 1] , f..l ( [0, aD = O? 4.10.24 Exercise. If f E MaN, then f* : JR. '3 x
H
limx f(y ) yt
�f f(x-)
exists and is in MaN and f* is left-continuous, i .e., f* (x) = lim f* ( y ) ; fur ytx thermore, f* � f. Similar results obtain for * f (x)
�f lim f(y ) �f f(x + ) . ytx
4.10.25 Exercise. If f E BV, then: a) U* , f * } c BV; b) there is a countable set 5 such that off 5, f* = f * ; c) for the jump function j(x)
�f L I f* (y) - f* (y) 1 y <x
associated with f, f - j E C(JR., q . [Hint: If f E MaN and f(x+) - f(x-)
>
0, then
Q n (f(x- ) , J(x+ ) ) :;to 0.]
4.10.26 Exercise. If 5
then f : JR. '3 x H
L
an '::; x
00
�f {an }N EN C JR., {jn }nEN C C, and L Un l < 00,
Un l i s in f E MaN and
* I f* (x) - f (x) 1
= { I jn l if x E � . otherwIse °
n= l
197
Section 4. 10. Miscellaneous Exercises
4.10.27 Exercise. If (JR., S)" p,) is a complex measure space and for x in JR., f(x) p,[(-oo, x)] , then: a) f E BV; x lim f(x) = 0; c) -
�f
t oo
f(x)
=
f(Y) ] . [�f lim ytx
f(x-)
d) The function f is continuous at x iff p,({x}) = O. (Properties a)-c) characterize functions of normalized bounded variation. The set of all such functions is NBV) .
4.10.28 Exercise. a) If f E JR.lR n NBV, then : Co(JR., JR.) '3 9 f--t is a DLS functional. The associated measure p, is totally finite and
I
Tf (x)
clef p, [(-oo, x) ] .
l
9
df
=
b) If f E ClR n NBV, there is a corresponding complex measure p, and
Tf (x)
�f 1p,1 [(-oo, x)].
4.10.29 Exercise. If f E BV( [a, b] ) , then varf is continuous at c in (a, b) iff f is continuous at c. 4.10.30 Exercise. a) If 9 E BV([O, 1] ) , for some a and b, -
g ([O, 1]) C [a, b] . b) If, for [a, b] as in a), f' E C([a, b] , JR.) , then f o g E BV([O, 1] ). *
*
*
Littlewood's Three Principles
In closing this discussion of real analysis there is an opportunity to mention some general guidelines [Lit ] that lie at the root of many of the arguments and ideas that have been presented. Real analysis began with Newton in 1665. In his time, a function was usually given by a formula and most formulre represented functions that were (at worst) piecewise differentiable. As real analysis grew and developed over the succeeding 300 years, there appeared functions defined by expressions of the form
f(x)
{A i f xES = � �;
B if x E T
xEU'
Chapter 4. More Measure Theory
198
The study of trigonometric series gave rise to highly discontinuous functions and led Cantor to discuss the sets of convergence and sets of divergence of the representing series. He turned his attention to set the ory itself and started an investigation that climaxed in 1963 with P. J. Cohen ' s resolution of the Continuum Hypothesis [Coh] . Other significant outgrowths of Cantor's work were general topology, Lebesgue's theory of measure, DLS functionals, abstract measure theory, probability theory, er godic theory, etc. The subject of functional analysis arose in an attempt to unify the methods of ordinary and partial differential equations and integral equa tions. The techniques were approximations that permitted modeling the equations by systems of finitely many linear equations in finitely many un knowns. In passing from the solutions of the approximating systems to what were intended to be solutions of the original equations, limiting processes were employed. At this point there appeared the need to conclude that the functions found in the limit were within the region of acceptable solutions. Therein lies the virtue of the completeness of the function spaces LP and the condition that a Banach space be norm-complete. (H. Weyl remarked that the completeness of L 2 is equivalent to the Fischer-Riesz Theorem. More generally, the norm-completeness of L 1 (hence of LP ) is essentially equivalent to the three basic theorems-Lebesgue's Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem- of integration.) Modern methods of TVSs relax the completeness con dition somewhat-topological completeness replaces norm-completeness but some effort is made to insure that limiting processes do not lead out of the spaces in which the solutions are sought. Topology gave rather general expression to the notions of nearness (neighborhoods) and continuity. Measure theory elaborated the notion of area since integration was for long viewed as the process of finding the area of a subset { (x, y) 0 :::; y :::; f(x) } of ffi? At the start, f was a continuous function and area was approximated by the areas of enclosed and enclosing rectilinear figures. The idea for this goes back first to Riemann, then Euler and Newton and ultimately Archimedes, who, using exhaustion, determined the areas and volumes of some nonrectilinear figures. With the advent of Lebesgue's view of integration, rectangle took on a new meaning, in the first instance, the Cartesian product E X [a, b] of a measurable subset E of ffi. and it closed interval [a, b] . Nevertheless, behind all the generalities lay the intuitive notion of the graph of a well-behaved function. When topology and measure theory were alloyed, the evidence became clear that many of the results were derivable by appeal to approximation of the general situations by others where the comfort of continuity and simplicity were available. :
Paraphrased, Littlewood's Three Principles read as follows.
Section 4. 10. Miscellaneous Exercises
199
1. Every measurable subset of ffi. is nearly the union of finitely many intervals. 2. Every measurable function is nearly continuous. 3. Every convergent sequence of measurable functions is nearly uniformly convergent. In the context of locally compact topological spaces and Borel measures the statement corresponding to 1. is: 1'. Every Borel set is nearly the finite union of compact sets.
4.10.31 Exercise. In what contexts is l' valid? 4.10.32 Exercise. Which of the results in Chapters 1 -4 exemplify Littlewood's Three Principles?
C O M PLEX ANALYSIS
5
Locally Holomorphic Functions
5 . 1 . Intro duction
In ClR the sets C k (JR., C), k = 0, 1, . . " satisfy the relations
Indeed, if fo
0,00) and for k in N,
= X[
/k (x) �f
{ oLXoo
/k- l (Y) dy if x 2': ° if x < °
then /k E C k (JR., JR.) \ C k+1 (JR., JR.) . Furthermore, for no x in JR. \ { a} is /k(x) representable by a power series of the form
00 cn (x - a) n (�f P(a, x). L n =O
On the other hand, as the developments in this Chapter show, if
f E (C'c , r > 0, and
z
z �f I' ( z )
f ( + h) - f ( ) lim h -> o h>'O
h
open
o
z in D(O, rt �f { z : z00E c, I z l < r }, the disc f dius r, then there is a power series L an zn �f P(a, z ) such that for all 00 z in D(O, rt, f ( z ) = L cn zn , i.e., the series in the right member conn=O verges if I z l < r and the sum is f ( z ) . Moreover, by abuse of notation, oo exists for all
fEC
ra -
(D(O, rt, q .
The striking contrast between the situations described in the two para graphs above is one of many motivations for the study of CIC . The principal 203
204
Chapter 5. Locally Holomorphic Functions
tool used in the investigation is complex integration-a special form of inte gration, for which Chapters 2 and 4 provide a helpful basis. The principal results from those Chapters are the THEOREMs about functions "( in BV
11
and associated Riemann-Stieltjes integrals of the form J d"(. In Section 1 . 2, fl represents a special ordinal number. Henceforth, unless the contrary is stated, the symbol fl, with or without affixes, is reserved for a region, i.e., a nonempty connected open subset of C regarded ffi.2 endowed with the Euclidean metric: as
d : ffi.2 3 [(a, b) , (c, d)] H J(a - c) 2 + (b - d) 2 E [0, (0).
Thus an element of C is regarded either as a pair (x, y) of real numbers or as a complex number z �f x + iy. When S e C, J E CS , and (x, y) E S, the notations J(x, y) and J(z) refer to the same complex number. Since polar coordinates r, fJ are frequently useful, when z = x + iy = re iO = a + re it , the notations J(r, fJ) and J (a + re it ) are also used for J(x, y) and J(z). For a complex number z �f a + ib, the complex conjugate of z is a - ib and is denoted z . This use of - conflicts with - used for closure in topology. Henceforth, for a subset S of C, S denotes the set of complex conjugates of the elements of S, whereas SC denotes the closure of S re garded as a subset of C endowed with its Euclidean topology. The term curve is reserved for a continuous map
"( : [0, 1] 3 t H "((t) E C;
the image of "( is "(* �f "(([0, 1]). When ,,( E BV([O, 1] ) , i.e., when "( is rectifi able, the integral t Jb(t)] d� (t) is sometimes' written J dz, or J dz, or
1
io
"I
1 1 J (a + re27l"it ) 27rire 27l"it dt.
5.1.1 ExerCise. a) For n E Z+-) J(z)
b) If J(z) = !, then
z
1
"I
=
n
z , and "((t)
J dz = 27ri.
=
e 21rit :
1
"I '
1 J dz
=
0.
Complex integration is used below to show that a function J differentiable in a region fl is locally representable by a power series: such that for some If a E fl, then C contains a sequence { } positive r(a) and all z for which I z - a l < r(a),
cn nEN
J(z) =
00
L cn (z - a) n . n=O
Section 5.2. Power Series
205
Thus the study of differentiable functions in CIC is reduced, in part, to the study, pursued with particular vigor and success by Weierstrafi, of power series convergent in some nonempty (open) discs of the form:
D(a, r �f { z : z E C, I z - a l < } . r
r
Later developments (v. Chapter 10) are concerned with methods of ex tending when possible, such functions to domains properly containing the disc within which the representing series converges.
(analytically continuing),
5 . 2 . Power Series
5.2. 1 THEOREM. (Cauchy-Hadamard) THE ( FORMAL ) POWER SERIES 00
Cn Zn � n=O
5 cle=f ,",
1 cle z = 0 OR I Z I < --==. 1 .--- =f Rs; 5 FAILS TO CONVERGE lim I cnl :;;: n ( 5 diverges) IF I z l > Rs.
CONVERGES IF
..... oo
1 1 [ 5.2.2 Note. The conventions = 00 and = 0 are observed. 00 o Hence the of 5 is in [0, 00] . For example, if = n!, n E N, = 00. It is occasionally = 0; if = n. convenient to denote by to emphasize the dependence of of the radius of convergence upon the sequence �f coefficients in 5.] -
-
radius of convergence Rs Rs Cn -\-' Rs Rs Rc
Cn
c
{cn } nEZ+
Rs :::; 00 and I z l < Rs, for some in (0, 1 ) , I z l nlim I Cnin < whence, for some s in ( 1 ) , all large N , and all k in N, r
PROOF. If 0
0, H [D(a, Rr] is substantial.
a
5.3.2 Exercise. a) If the radius of convergence of
in D(O, Rr, J(z)
7ri
00
�f L
00
L cn zn is R and, for z
n=O n Cn z , then J E H [D(O, Rr]. b) If 0 < r < R and
n=O 2 t l'(t) = re , 0 -::; t -::; 1 , then
i J(z) dz
=
O.
[Hint: a) Induction yields the identity w n - zn n - l = (w - z) (w n - 2 + 2zw n - 3 + · · · + (n - 1)z n - 2 ) . w - z - nz Thus when max { I z l , I z + h i } -::; r < R and I h l is positive,
---
I
J(Z + h) - J(Z) h = Ihl
-::;
l�
-
� ncn z n - 1 L-t
n= l
1
Cn [ (z + h t - 2 + 2z(z + ht - 3 + . . . + (n
� f n(n - 1 ) l en I rn - 2 .
I I
- 1)zn-2] I
n= 2
Then 5.2.3 applies. b) 5.2.4 applies.]
5.3.3 Exercise. If l' is a rectifiable closed curve, g E C ( 1' * , C) , and, for w) z not in 1' * , J (z) �f g ( dw, each component C of C \ 1' * is a region
and J E H (C).
1
-y w - z
Chapter 5. Locally Holomorphic Functions
210
5.3.4 THEOREM. IF 1 E H (Q) ,
I(z) �f I(x + iy) �f u(x , y) + iv(x, y) [�f �(f) + i CS (f)], AND z E Q, THEN THE Cauchy-Riemann equations
ux (x, y) = V y (x, y) and uy (x, y) = -vx (x, y) OBTAIN. PROOF. If z = x + iy, then !' ( z) = lim h�O h ";O
1 ( z + h)
h
-
1 (z)
= Ux + ivx I(z) = lim I(z + ih) ih =
-
h�O h ";O
-
o
iuy + v y .
[ 5.3.5 Remark. In terms of the operators
(�
a �f � �f � az 2 ax
_
)
i� ' ay
a �f
� �f � az
(�
+ i� 2 ax ay
a the Cauchy-Riemann equations reduce to -a1 = azl
=
)
o. m
In this book the same symbol a appears in different contexts: au analysis, - , the partial derivative of u with respect to x; in topolax ogy, a(U) , the (topological) boundary of U ; in complex analysis, al as introduced above. Nevertheless, the intended meaning of a whenever it occurs, is clear.
a al . . 1 denvatIve · . · · fy t he partIa The" notatlOn resp. l does not slgm az az of 1 with respect z resp. z. The alternative notation a1 resp. a1 is less likely to be misinterpreted.] -=
5.3.6 Exercise. If: a) Q is a region in C; b) u and v are in ffi.r1 ; c) the derivative of the map T Q 3 (x, y) r-+ [u(x, y), v (x, y)] E ffi.2 exists; d) in Q, the Cauchy-Riemann equations obtain, i.e., Ux = Vy and uy = Vx , then for some 1 in H(Q), I(x + iy) = u(x, y) + iv(x, y). (If partial differentiability of u and v only off a countable set and the Cauchy-Riemann equations only :
-
Section 5.3. Basic Holomorphy
211
a.e. in Q are assumed, a result of Looman and Menchoff implies nevertheless that J E H(Q) , v. [Sak] , (5.11), p.199.) The next development leads to the connection between differentiability, i.e., holomorphy, of a function J in en and, for each D(a, rt contained in Q, the representability of J (z) as a power series P( a, z) ( converging at each point z of D(a, rt). The fundamental tools are Cauchy 's Integral Theorems and Cauchy 's Jormul(£;.
5.3.7 Exercise. a) If {A, B, C } c e and a(ABC) is the 2-simplex deter mined by A, B, and C,
1
8[a(ABC)] b ) If
/,(t) =
=
[A, B] U [B, C] U [C, A] . 1
. 0 0 and R is if R i- z, I(z) R-z R-z near but different from z, owing to the continuity of I, =
Ir
J[Z, R]
l
[J(w) - I(z)] dw � E I R - z I .
J
'Y
o
5.3.16 COROLLARY. IF Q IS convex, IS A RECTIFIABLE CLOSED CURVE, AND C Q, FOR 1 AS IN 5.3.15, 1 dz = O.
'Y*
'Y
"i
PROOF. If is piecewise linear, since
(.1)' = I, FTC implies
i 1 dz = 1 1 (.1) ' h(t)] 'Y'(t) dt =
1 1 [10 'Y(t) ] , dt = 1 0 'Y I � .
215
Section 5.3. Basic Holomorphy
Since ,,/ is a closed curve J 0 "/ I � = O. If "/ is a rectifiable closed curve, as in the last part of the PROOF of 5.2.5, there is a sequence {"/Pn } nEN such that lim 1 dz = 1 dz. n-+= 'Y n Each integral in the left member above is zero. 0 5.3.17 Exercise. If Q is convex, {P, Q } c Q, and "/1 , "/2 are two rectifiable curves such that:
J
a) b)
J
"I
P
"/� U "/� C Q, "/1 ( 0) = "/2 (0) = P, "/1 (1) = "/2 (1) = Q,
J"11 1 dz = J"12 1 dz: the integral is independent of the path. If, in 5.3.15, the hypothesis that Q is convex is dropped, J 1 dz can then for any 1 in H (Q),
"I
be different from o.
5.3.18 Example. If Q cle=f C \ { 0 } and 1 ( z) = - , then 1 E H (Q), and Q IS. Z not convex. If "/ ( t) = e 27rit , 0 :::; t :::; 1, then "/ * C Q and
1
On the other hand, since Ind 'Y (O) = 27ri, the presence of 27ri in the right member above suggests the possibility of a formula that relates Ind (0) and some integral involving I. In the derivation of the formula, the importance of the hypothesis 1 E C(Q, q n H (Q \ P } ) in 5.3.11 becomes clear. "I
{
5.3.19 THEOREM. (Cauchy's integral formula, basic version) IF Q IS A CONVEX REGION, "/ IS A RECTIFIABLE CLOSED CURVE SUCH THAT "/ * C Q AND 1 E H(Q) , FOR EACH a IN Q \ ,,/* ,
1
I(a) · Ind 'Y (a) = -. 27rl
J I(z)a dz. "I
--
Z
-
(5.3.20)
[ 5.3.21 Note. If Ind 'Y (a) = 1 and g(z) �f I(z)(z - a) , (5.3.20) yields I(z) dz = g(a) = 0, i.e., a significant generalization of 5.3.11. The larger message of (5.3.20) is that the value of 1
i
at any point a in Q is a complex weighted average of the values of 1 on "/* . Mere continuity of 1 is insufficient for such a conclusion since, absent the assumption of the differentiabilty of I, 1.2.41 implies that for some continuous I, f(a)=l while 1 1 "1 ' = 0.]
216
Chapter 5. Locally Holomorphic Functions
PROOF. Since J(a)Ind ,),(a) = ity of
dz = o. 1')' J(z)z -- J(a) a
� 27rl
1')' zJ(a)- a dz it suffices to prove the valid-
To this end 5.3.16 is applicable because, by definition,
{
= Fa : (""\ 3 z clef H
J(z) - J(a) z-a f' (a)
if z ¥= a if z = a
(5.3.22)
is in C(Q, C) n H (Q \ {a}) and thus 5.3.16 implies (5.3.20).
o
5.3.23 COROLLARY. IF a E Q, D(a, r ) c Q, AND J E H (Q), FOR SOME SEQ U ENC E { cn (a) } nEZ + IN C AND ALL z IN D(a, r t ,
J(z) =
00
L Cn (a)(z - a) n .
( 5.3.24)
n=O
PROOF. For "( : [0 , 1 ] 3 t f-t a + re 27rit , Cauchy's formula and 5.2.11 imply that if z E D(a, r t , then
(1 Fz (w) dw 1 J(z)- d ) = �27rl (0 1')' J(z)- d ) J (w) = _1 dw . 27ri ')' (w _ a) ( _ z - a )
J(z) =
Since
� 27rl 1
I I z-a w-a
+
')'
1
< 1
')' W
w-a
for all w on "(* ,
Z
+
W
1
f (� � : )
= 1 _ z - a n=O w-a
counting measure, Fubini's Theorem applied to
W
n
Z
W
v
. When is
justifies the subsequent interchange of integration and summation:
Thus cn (a) =
� 27rl
1(
j +1
J (w
dw . -an 5.3.25 Exercise. If ( X , S, f.-l ) is a complex measure space, ')'
W
g E 5, U E O(C), and g(X) n U = 0,
o
Section 5.3. Basic Holomorphy
then
f
:
U3
z
r-+
z
217
rx g�xp,)(x)- z is in H (U).
i
a
[Hint: If E U, for some in U and some positive
z E D(a,rt c U. z - a :::; -I z - al < 1 and For each x in X , I r g (x) - z I
r,
(z - a) nn . g(x) - a (g(x) - a) ( 1 - z - a ) n=O (g(x) - a) + l g(x) - a 1
1
Again, when
v
(Xl
=L
is counting measure, Fubini ' s Theorem applies to
. J g (dp,)(x-) = �� Jx (g(x(z) -- a)a)nn+ 1 '] n=O m
and valIdates
X
X
Z
f( ) exists and f(rn) (z) = n=L n(n - 1) · · · (n - m + l)cn (a)(z - a) n -rn rn (whence f ( rn ) E H(Q) ) . In particular, 1 f(w)n + dw. (5.3.27) Cn (a ) = f(nn!) (a) = 27ri J'Y (w - a) l Furthermore the radii of convergence of the series representing f and f ( rn ) are the same. 5.3.28 Exercise. If M(a, r) �f max I f (w)1 �f max I f (a + re 2 7l"it ) I , O�t � l I w - a l =r 1 then M(a, r) 2 z 1 I f (a + re27l"it ) 1 2 dt and 5.3.26 Exercise. In the context of 5.3.23, if m E N, 00
00
2 r 2n . c i (a)1 n n=O [Hint: Fubini's Theorem applies; {e 27l" nit } nEZ is ON on [0, 1] .] =L
Chapter 5. Locally Holomorphic Functions
218
The inequality
00
L I cn (a)1 2 r2n � M(a, r ) 2
n=O
Gutzmer's coefficient estimate: which implies the (weaker) Cauchy estin! M (a, r) mate.. I f (a ) 1 - r A function in H (C) is an entire function. The set of entire functions is
is
d, a con b D( a, rt, w = f(b), D(f(a) , dt c f (D(a, rn [ 5.3.40 Note. The PROOF above of the Open Mapping Theorem resorted to the Minimum Modulus Theorem, a consequence of the
i.e., tradiction. Hence, for some in
0
Maximum Modulus Theorem:
{ Maximum Modulus Theorem } ::::} { Open Mapping Theorem } . Conversely, the Open Mapping Theorem implies that a non constant function in maps onto a region: if E for some positive the open set is contained in and for some positive C If = each ion satisfies = O. If the half-line starting at and passing through goes on to meet in a point such that
f H(Q) Q a Q, r, f (D(a, rt) f(Q) s, D(f(a), s) f(D(a,rn. f(a) f(a)0, 0, f(b) 8[D (f(a), stJ 0 If(b)1 > I f (a)1 f ( a) 8[D (f(a), srJ f(b) I f (b)1 > I f (a)l: { Open Mapping Theorem } ::::} { Maximum Modulus Theorem } . J
Chapter 5. Locally Holomorphic Functions
222
J
a E Q, AND J - 1 1 J[D(a,r) O]
5.3.41 THEOREM . (Inverse Function Theorem) IF E H (Q) , IS INJECTIVE AND i- 0, FOR SOME POSITIVE r,
J'(a)
JI D(a,r) O
IS HOLOMORPHIC. PROOF. Since i- 0, for all in some
J'(a)
z
D(a, rt, J'(z) i- O. J(w) - J(z) if z i- w G : Q x Q (z, w) r-+ { I'(z) w-z if z = w which is in C (Q2 \ { (z, w) : z = w }) is shown next to be in C ( Q x Q, q . Since I' E H (Q) , I' is continuous. If z is near but different from w , and l'(t) �f (1 - t)z + tw, then 1'(0) = z, 1'(1) = w, 1" (t) = w - z, G(z, w) - G(z, z) = w 1 z 10 1 {Jh(t)]}' dt - G(z , z), 1 = 1 [I' h (t)] - I'(z)] dt, whence I G (z, w) - G(z, z)1 is small: G E C ( Q x Q, q . Thus, for some positive < r and if max{ l z - ai , I w - al } < then I G(z,w)1 � IJ'(a)1 > 0, 1 i.e. , if {P, Q } C D(a, ) then IJ(P) - J(Q)I 2 1 1'(a)I ' I P - Q I . Hence J I D(a.s) O is injective: for some g defined on J (D(a, t ) , g o J(z) z. If b E Q 1 and Q 1 w i- b, for some P,Q in D(a, t , P-Q g(w) - g( ? ) (5.3.42) w-b J(P) - J(Q) " Since Q E D(a t , J'(Q) i- o. As w -+ b, P -+ Q and the right member of 1 (5.3.42) converges to I' (Q) ' [ 5.3.43 Note. If 0 = a = J(a), w = J(z) = L an Zn , 0 :::; Izl < r, a 1 i- 0, n=1 and when I w l is small, z g(w) = m=1 (5.3.44) L bm wm . The function
3
_
s,
s, s
�
s ,
�
3
,s
s
s
==
0
00
=
00
Section 5.3. Basic Holomorphy
223
Thus
Comparison of coefficients leads to a sequence
(recursively)
cal of formulre from which the sequence {bn } n EN is culable in terms of the sequence {an } n EN . (Similar but more com plicated formulre obtain when a and I(a) are more general.) The Cauchy estimates imply for M �f M(a, r),
If lzl < r, the series l a 1 I z -
oo
M n M 1 z = l a 1 I z - -2 z 2 -Lz reprer rn 1--
n =2 r sents a function F, a of I, i.e., a function F for which the power series coefficients in absolute value the power series coefficients for I. The equation W = F(z) is quadratic in z, and if
majorization majorize
then G o F(z) = z. At and near 0,
whence, for some positive p depending only on 11'(0) 1 r and M, G' exists in D(O, p t . There is a sequence {cn } n EN such that for W in D(O, pt , G(W) = Cn Wn ( = z). The recursive formulre n=1 for the sequence {cn }n EN show Ibn l :::; Icn l , n E N. Thus the series (5.3.44) converges if Iwl < p: 00
L
1 (D(O, r n ::J D(O, pt · J 5.3.45 Example. If I(z) = then 1 E though 1 is it is not
E,
= I, and f' is never zero. Al locally injective eZ, globallyI'injective: for n in Z, e 2mri = 1.
224
0 lal, 0
Chapter 5. Locally Holomorphic Functions
5.3.46 Exercise. If < < for some L in H ( D (a , rr ) , eL ( z ) = z. [Hint: The function L in 2.4.21 serves.] The condition !, (a) i- plays a central role in the PROOF of the major part of the Open Mapping Theorem, 5.3.39. There is a refinement that deals with the circumstance: 1 is not a constant function, but !, (a) = O. r
5.3.47 THEOREM. I F 1 E H(Q) , a IS IN Q, AND 1 IS NOT A CONSTANT FUNCTION, FOR SOME m IN N, SOME NEIGHBORHOOD N(a) , A g IN H [N(a)] ' AND ALL z IN N(a) I(z) = I(a) + [g(z)] Tn . FURTHERMORE, FOR SOME b AND A POSITIVE g[N(a)] = D ( b, r �f V, g I ( a ) ' IS INJE CTIVE, N AND FOR SOME h IN H(V) , h 0 g I ( a ) (z) == z. r,
r
N PROOF. For some m in N and some N(a), if z E N(a), then I(z) = I(a) + (z - a) Tn
[� a)n1 �f cn (z -
I(a) + (z - a) Tn k(z)
k' k' and k I ( a ) i- O. Hence k E H [N(a)] ' and for some h, h' = k ' If z E N(a) , N then {k(z) . exp[-h(z)] } ' = exp[-h] [k' - kh'] = O. Hence, for some con'0 , k exp (-h) = stant de=f on N(a). Since Tn h k= exp ,
M I M l e"
[I M I ,!;
M
(
:iO )]
0
if g (z ) �f (z - a)k(z) , then I(z) = I(a) + [g(z)]Tn. Moreover, g ( a ) = and g' ( a ) = k(a) i- o. 0
0,
5.3.48 THEOREM. (Morera) IF U E O(q , a E U, 1 E C(U, q , AND FOR EACH 2-SIMPLEX CONTAINED IN U \ {a}, r 1 dz = THEN 1 E H (U). a
Ja(a) [ 5.3.49 Note. The open set U need not be connected, e.g., the conclusion is valid if U �f D(O, lrl.,JD (3, lr �f l!Jl.,JD(3, l r .] r
0
r
PROOF. If > and w E D(b, r c U, the hypothesis implies (even when r 1 dz unambiguously defines an F in b = a) that the formula F( w) J [b , wl cD( b , r)o . Moreover, FTC implies F' exists throughout D(b, rr and F' = I. Hence F E H (D(b, n and thus F' (= f) E H (D(a, r ) 0 5.3.50 Example. The hypothesis 1 E C(U, q in 5.3.48 cannot be omit
�
r
,
r
ted. Indeed, if
1 ( z) =
{ 0:2
0
if z iotherwise
·
Section 5.3. Basic Holomorphy
and
a
225
is 2-simplex contained in C \
r I dz = o. Nevertheless {O}, then lara)
there is no entire function g such that g l lC\ {o} = I. Another aspect of holomorphy is highlighted by contrast in
1 ( ( ))k
5.3.51 Example. For each k in N, the function
j" ll!. " X h
Ik ( � )
27r SIn ' 2
;
0
if x
i- 0
otherwise
i-
is continuous on JR., 0 , k, n E N, and if k l, h and II are different functions: Two (different!) functions in C(JR., JR.) can assume the same values on an infinite set, e.g., �f , such that 0.
S
=
{.!.} nEN
S· i-
n
The following result, which is of general importance in the context of holomorphic functions, provides the contrast to 5.3.51. 5.3.52 THEOREM. (Identity Theorem) IF I E H(Q) ,
S �f {adk EN C Q,
and Q
::)
S· i- 0,
THE VALUE OF I(z) IS DETERMINED FOR ALL z IN Q BY THE VALUES I PROOF. By hypothesis, for some in and some subsequence =I contained in Q, lim = If g E H (Q) and g n E N, for
{ (ak )}kEN ·
a S· {an }nEN ) a a. (a (a ) , n n n n-+= some positive I - g �f h is representable by a power series in D( a, t and h(an ) = 0, n E N. If h i- 0, for some M in N and all z in D (a , t , r,
r
an are in D( a, ) 0 , for all large n, 0 = h (an ) = (an - a)M (� Crn (an - a) rn ) . Since a n - a i- 0, o = for all large n, Co + L Crn (an - a) rn = o. The second term in the left rn= and Co
i- o.
r
But since all but finitely many
1
r
Chapter 5. Locally Holomorphic Functions
226
member above converges to zero as a n
I
h D(a,r o = 0 ) If Q = D( a, r t
--->
a,
whence
Co
=
0, a contradiction:
the argument is complete. If b E Q \ D( a, r t there is a set {[ zn , zn + dh �n� N of complex intervals such that Z 1 = a, ZN = b, and P �f
N
U [zn, zn+ d c Q (v.
n= 1
1.7.11). If w E P, for some positive r(w),
h
is representable for all Z in D[w, r(w)]O by a power series. For each r, the open cover {D[w, r(w)]O} wEP of the compact set P , admits a corresponding finite subcover: { D (wi , ri t L < i
ex:>
233
Section 5.4. Singularities
a
5.4.9 THEOREM. IF 1 E H(Q) , AND IS A ZERO OF ORDER no, THEN FOR SOME NEIGHBORHOOD ( ) , WHENEVER b E I( ( ) \ { } )
Na
i.e., 1 IS AN no-to-1 MAP OF
Na a ,
N (a) \ { a} .
if E D( , r �f
a N1 ( a ) , then I ( z ) = ( z - a ) no L cn ( z - a t �f ( z - a ) no g ( z ) n=O and if z E N1 (a), g ( z ) i- O. Thus !L E H [N1 (a)]' and since N1 (a) is convex, g g g' '(() if, for z in N1 (a), G(z) = -- de then G E H [ N1 (a)] and G' = - . g [a, z ] g ( () PROOF. For some positive
r,
z
r
00
,
1 ' G (Jl...- ) - e g' - g (eG) ' e G g' - ge G G' = 0 and for some constant Thus G e 2G 2G K K + K, g = : G . It foU:ws that if h( z ) = ( z a ) exp ( � G ) , then _
_
K+G(a)
h'(a) = e-n-o
- i- 0, the argument in 5.3.41 shows that h is injective Since on some open subneighborhood ( ) of ( ) and that h ( ) ] is open. Hence, for some t in (0, 1),
n
s
If 0 i- b E D (0, t O t, for some and some
={
If S clef
Wk
no exp [i ( fJ +n2k7r )]
clef ...L
=
S
[N2 a
N2 a N1 a
w . . . , wno -
O
fJ,
and # ( { o, d ) = no. Since h is injective on injective on the (open) neighborhood
Hence
}
: 0 :::; k :::; no - 1 , then
N2 (a), h is also
N ( a ) \ { a } contains precisely no points Zo, . . . , zno - 1 such that
234
Chapter 5. Locally Holomorphic Functions
D of
b, a
H(Q)
no N(a): ¢ H[N(a)]; b + ¢no; ¢ ¢' N(a). 5.4. 11 COROLLARY. a) IF a E Q, I E H(Q \ {a}), AND a IS A POLE OF ORDER no OF I, FOR SOME POSITIVE R, {I b l > R} :::;. {# [J - l (b) n Q] = no} . b) FOR SOME POSITIVE r AND AN INJECTIVE ¢ IN H [D( a, ) ] , I = ¢nu . PROOF. a) If g(z) �f (z - a) no / (z), 5.3.41 implies that for some positive s, g is injective on D(a,sr · Then I I D( a, s) O ' which is z (z g(z)a) no ' . an no-fold map of D(a, s)O on { z I z l > � �f R } . 5.4.10 Exercise. If I E and for some is a zero of order a) E 1 - for some function and some neighborhood b) 1 = c) is injective and is never zero in
b,
¢
r
H
b) 5.4.10 applies. 5.4.12 Exercise. If I E 1- 1 E 5.4.9 applies.]
°
_
IS
:
D and l i n is injective, f' l n is never zero and
H(Q) H[/ (Q)]. [Hint: 5.4.13 Exercise. If I (z) = eZ, then: a) for each in C, 1'( 0' ) -j. 0; b) for some positi ve r, II D ( ,r ) O is injective; c) for the g in the PROOF of 5.4.11, g l f( D ( ,r) o ) E H (Q d; d) g (Q d c U Ln ( w ) ; e) g' (w ) = �; f) if h E H(Q), wEn, aH E[D(Q, ,rr and h( a) -j. 0, for some positive r, D( a, r r c Q and for some L in a ] ' h(z) I D(a ,r) o = eL (z) . If r > 0 and I E H [D(a, rr]' the basic version of Cauchy's integral formula (5.3.20) leads to (5.3.24), i.e., the representation of I by a power series converging to I throughout D( a, r r. A global version of Cauchy's formula leads to a similar representation when a is an isolated singularity of I. If a is not a removable singularity, the representation of I cannot converge at a but at best in the region D( a, r r \ { a} . A generalization of this kind of region is an annulus, i.e., when 0 :::; r < R < 00 , a region of the form { z : r < I z - al < R} �f A( a; r, Rr (for which the closure is A( a; r, R) �f { z : r :::; I z - al :::; R}) . An annulus A( a; 0, R) resp. open annulus A( a; 0, Rr is a punctured disc resp. a punctured open disc at a and is denoted D( a, R) resp. D( a, Rr. (For a given annulus A( a ; r, R) when 0 :::; fJ < ¢ :::; 27r the open annular sector is A(a : r,R;fJ,¢r �f { Z : z = a + pei,p , r < p < R, fJ < '1/J < ¢}.) a
235
Section 5.4. Singularities
A reasonable approach to such a representation for an J holomorphic in A( involves, when < < < an integration over the two curves, and such that
a; Rr, r,
'Yl
(The set
s S R,
r
'Y2
'Y � U 'Y� is the boundary
8 [A(a; s,srl = { z : I z - al = s } u { z : I z - al = S} .) For any w not in A( a; Rr, direct calculation shows that r,
Ind 'Y! (w) + Ind 'Yz (w)
=
o.
These remarks motivate the following 5.4.14 THEOREM. (Cauchy's integral formula, global version) IF U IS AN OPEN SUBSET OF C, J E H ( U ) , ARE RECTIFIABLE CLOSED
CURVES,
{-ydl :s;k «,K K U 'YZ C U, AND FOR EACH w NOT IN U,
k=l
FOR EACH
K
2:)nd 'Yk (w)
K
a IN U \ U 'YZ , J(a)
k=l
=
(5.4.15)
0,
k=l [2:) 1 = LK � dz. 2�l k= l J - a k= l K
nd 'Yk (a)
1
J( )
-.
'Yk
�
{ w : t Ind 'Yk (w) = o } is open, contains C \ U,
PROOF. Because the left member of (5.4. 15) is z:.-valued and depends con tinuously on w, V �f and so V U U = C. For the function
G : U x U '3
k=l
(z, w)
H
{ J(w)w - zJ(z) if z -j. w I'(z)
_
if = w
z
Chapter 5. Locally Holomorphic Functions
236
introduced in the PROOF of the Inverse Function Theorem, (5.3.41), the hypothesis (5.4.15) implies that the formulre
G(z) dgf
1
K
LJ
G(z,w) dw if z E U t (w Z du if z E V k= i 'Yk � �
are consistent on U n V: G is well-defined throughout C. Since G E H (U), E . Because
C E H(V) , and U n V =I- 0, it follows that C E K V ::) n Ind 'Yki ( O ) , k=i { z : I zl
I z l , I C (z)1
> R }. For large for some positive R, V ::) is small and thus 5.3.29 implies = 0. The promised conclusion follows when the equation == ° is written in terms of the defining formulre for 0 [ 5.4.16 Note. The hypothesis (5.4.15) is satisfied if, e.g., as in Figure 5.4.1, for some positive R,
C(z)
C
C.
{ad i �k � K C D(O, Rr , K 27r t , 1] , 1 :::; k :::; K, and U 'YZ C D(O, Rr · 'Yk (t ) = a k + rk e i , t E [0 k=i U = D(O, Rr ,
In Figure 5.4.1, the dashed lines together with the small circles themselves may be construed as the image of a single rectifiable
Figure 5.4.1
Section 5.4. Singularities
237
closed curve, say r. Integration over r can be performed so that the integrations over each dashed line are performed twice (once in each direction) with the net effect that those integrations contribute nothing to The validity of 5.4.14 for the configuration just described follows directly from the basic version of Cauchy's Theorem. The approach in 5.4.14 permits a very general result free from appeals to geometric intuition, v. Figure 5.4.l .
1.
]
5.4.17 Exercise. In the context above, K
K
k=L1
1 J dz 'Yk
= o.
[Hint: If a E U \ k=U 'YZ and 1(z) �f (z - a)J(z) Cauchy's formula 1 applies to T]
bk} �= 1 and { Jj};'= l are two sets of rectifiable closed J curves such that U 'YZ U U J; c U and for each w not in U, k= 1 j = 1 5.4.18 Exercise. If K
J
K
j=1
k= 1
L Ind 8; (w) = L 1nd 'Yk (w),
1 J dz k=L1 1 J dz. jL= 1 J
then
8;
[Hint:
=
K
'Yk 5.4. 17 applies to the calculation of J
K
fdz - k=Ll'Yk Jdz.] jL= 1 l8) 1 The following is a useful consequence of 5.4.14.
z A(a; r, Rt, J E H [A(a; r, Rt]' FOR SOME {cn (a)} nEZ IN C AND
5.4.19 THEOREM. IF ALL IN
00
n= - (X)
(5.4.20)
Laurent series
J
THE RIGHT MEMBER OF (5.4.20) , THE FOR IN THE AN NULUS A( CONVERGES UNIFORMLY ON EACH COMPACT SUBSET OF A( R) 0
a; r,
a; r, Rt, •
Chapter 5. Locally Holomorphic Functions
238
PROOF. If Z E A(
a;
r,
Rr, for some [s, S] contained in ( R) , r,
A(a; s, Sr. If 'Yl (t ) = a + se 2 7ri( 1 - t ) and 2 (t ) = a + Se 2 7rit , 5.4.14 implies J (w) J (w) dW J(z) = � 27rl [1 W - Z dw + 1 W - Z ] . For w in the first resp. second integral of (5.4.21), I wz -- aa I < 1 resp. I � w - a I < 1. Hence � (w - a) n 1 1 . 1 = � (z - a) n + l w, ---a-:- n=O w - z z --a -----:1 - -z-a resp. � ( z - a) n . 1 1 1 � z a w - z w - a 1 w - a n=O (w - a) n+ l If s < t < T < S, both series converge uniformly in A( a; t, T) . If -� J (w ) (w - a) n dw if - 1 27rl 1 Cn ( a) = -1 J(w) dw if 2': 0 27ri J ( w - a ) n + l Z
1'
E
')'2
')'1
-
_
{
(5.4.21 )
n
')'1
n
')'2
:::;
(5.4.20) obtains. D is not assumed to be 5.4.22 Remark. Owing to the fact that [
a, J
0 [ a E Z (f)] if n < 0 [ a E P(f)] .
241
Section 5.4. Singularities
{m,n } C ;2;+ , I (z) = (z - a) Tn + ( z -1 b) n , ,,( is a rectifiable closed curve such that { a, b} and 0 < 1 z - bl < 1 a - bl , then: bI) the Laurent series for 1 takes the form b) If
r:t.
* "( ,
{dp }pEZ
j
in the Laurent series for can be calculated b2) the coefficients by comparison of coefficients of like powers of in the two members of
z-b
Tn k - b) k l -�n l + L _ (z b + k= l Ck (Z -
Cz � b)n � Ck (Z - b) k) . Ct dp (z - b)P) .
=
+
A similar calculation when 0
. The formula is known as the or the It is most useful when
{f b(t)]} 271" Argument Principle .
Principle of the Argument
"((t) re2rrit , t
For example, if Q = C and = 0 ':::; .:::; 1 , the formula provides the difference between the sum of the orders of the zeros of in and the sum of the orders of the poles of in
f D(O, rt D(O,rt· A detailed discussion is given in 5.4.37.]
PROOF. At each point meromorphic function
f
a in Z(f) resp. b in P(f), the Laurent series for the j takes the form
The Residue Theorem 5.4.28 applies. D 5.4.35 Exercise. If E H(Q) , E Q, and i- 0, for some positive �f Q I , and for some L in H (Qd then c Q, ° tJ. Furthermore, if "( is a rectifiable curve and = f.and (L I)' = * "( C then L "( E 1] and
f a f(a) r, D(a,rt f(D(a,rt) e Lof j. D(a,rt, f BV([O , ) 1 ff' dz = 1 1 d[L f (t)] (Riemann-Stieltjes integral!). [Hint: 4.10.30 applies.] 0
'Y
0
°
0
0
0
"(
5.4.36 Exercise. Under the hypotheses of 5.4.33, if g E H(Q) , then
1 f " g( a)Ord f( a )Ind ( a) � 271"i 1 g-f' dz = aE(Z(J)nD(O,r)O ) g ( b)Ord f( b)Ind 'Y(b). b
-
'Y
'Y
E ( p(J)nD(O,r)O)
[Hint: The Laurent series for g j at the points in Z(f) and P(f)
are useful.]
Section 5.4. Singularities
243
5.4.37 THEOREM. (Argument Principle) IF: a) I E M (Q); b) "( IS A REC TIFIABLE CLOSED CURVE; AND c) "( * c {Q \ [Z(f) u p(fm , THEN:
= r IS A RECTIFIABLE CLOSED CURVE; A ) I 0 "( clef B) ind r (O)
=
b E [p(J)nfl]
aE[Z(J)nfl]
[ 5.4.38 Remark. The left member of the formula in b) is the winding number of the curve 1 0 "( about O. The result is most useful when, for each c in Z(f) U P(f), Ind ,,(c) 1.] =
11
PROOF. a) Since 1 1' 1 is bounded on stant M, I r(s) - r (r ) 1
=
h(r) ,,,(s)]
* "( ,
if 0 .:s; r < s .:s; 1 , for some con
l
I' (z) dz ::; M h(s)
- "((r) l , v. 4.8.6.
Because "( is rectifiable, r is rectifiable; because "( is closed, r is closed. b) By virtue of 5.4.35 and the compactness of "( * , for some Riemann part ition {tk L � k�n of [0, 1] and positive numbers {rd l � k � n : bl) "(
*
n
c
U D b (tk ) , rk t ;
k= l
b2) D b (tk ) , rkt n D b (tk + l ) , rk +l t -j. 0, 1 .:s; k .:s; n - 1; b3) there is an £k in H {D b (tk) , rkt } and such that e £ k o j = I ; b4) £k + l 0 1 - £k 0 I is constant on D b (tk) , rkt n D b (t k + d , rk + lt, and in 27ri . Z. Finally, 5.4.28 applies. D
}
5.4.39 COROLLARY. (Hurwitz) IF { In n E N C H(Q), 0 tJIn � I
U In (Q) ,
nEN
AND
ON EACH COMPACT SUBSET OF Q, EITHER 1 == 0 OR 0 tJ- I (Q ) . PROOF. If I t o, since I E H (Q), Z(ft n Q 0. Hence, if E Q, for Furthermore, 0, some positive r, 0 tJ- J [ 0, r r] and E H
1 n
A( a;
1
j
[ A (a; � fJ . =
a
fn dz = -. lim -. - dz. The left member above is 27rl Iz - a l = � f 27rl I z -a l = � I zero, whence so is the right. If I t O , for each in Q, I is not zero in some neighborhood of n--+=
1
o tJ- I(Q).
a
1
1'
a:
D
Chapter 5. Locally Holomorphic Functions
244
5.4.40 THEOREM. (RoucM) IF { f, g} c H (Q) , D(a, r) c Q, AND
(5.4.41 ) THE SUMS OF THE ORDERS OF THE ZEROS OF f AND OF f + g IN D( a, r t ARE THE SAME: aEZ(f+g )nD(a,1') O
aEZ(f)nD(a,r)O
PROOF. For t in [0, 1] ' the hypothesis (5.4.41) implies that the integral 1 (f + tg)' �f N ( t ) dz -Iz -a l =r f + tg
27ri 1
is well-defined. According to 5.4.33, N(t) is Z-valued. On the other hand, the left member above is a continuous function of t and must be a constant. Moreover, N(O) = aEZ(f)nD(a,r)O
N(l)
=
aEZ(f+g )nD(a,r)O
and, since N is a constant function, N(O) = N(l). D 5.4.42 Exercise. If, for the polynomials p, q, deg(p) = M < N = deg(q) and R is sufficiently large, then { I z l 2: R} :::;.. { lp(z) 1 < Iq(z) I } · 5.4.43 Exercise. If aN i- 0,
f( z ) cl=e f
N""- l n=O �
NN N
n
a z , g ( z ) cl=ef a z , a n
--Ir
0 , and h = f + g,
5.4.40 and 5.4.42 imply that for R sufficiently large,
L aEZ(h)nD(a,R)O
Ord h(a)
=
N,
i.e., the strong form of FTA is valid: if p is a polynomial of degree N and multiplicities of zeros are taken into account, p has N zeros. 5.4.44 Exercise. a) If a is a simple pole of f, in some nonempty open N( a) \ {a}, f is injective. b) If a is a simple pole of both f and g, some linear combination h �f of + (3g is holomorphic in some nonempty open neighborhood of a.
Section 5.5. Homotopy, Homology, and Holomorphy
245
5 . 5 . Homotopy, Homology, and Holomorphy
The close connection between ind 'Y and Ind 'Y when "( is a rectifiable closed curve suggests that there is a topological basis for many of the results about complex integration. An approach that reveals this basis is found in the next paragraphs. The fundamental material about homotopy is given in Section 1 .4. 5.5.1 DEFINITION. A CLOSED CURVE "( : [0 , 1] H Y IS null homotopic in A IFF FOR SOME CONSTANT CURVE 15 : [0 , 1] '3 t H r5(t) == y E Y: "( AND 15 ARE homotopic in A. 5.5.2 DEFINITION. FOR TWO CURVES "( AND 15 IN A TOPOLOGICAL SPACE Y, WHEN "((I) = 15(0) THE product "(15 IS THE CURVE
"(15 : [0 , 1] '3 t H
{
1
"((2t)
< t 0, for some polygon 7r* and some homotopy F, "I F, n 7r and 0 = Ind 1J(O) =
'!L dt = 1]
0
0
rv
t Esup[O,! ] h(t) - 7r(t) 1 < E. b) If "I and 15 are rectifiable curves, "1* U 15* c n, and for some homotopy F, "I F, n 15, there is a finite sequence {7rZ L R for near but not equal to a pole of I, I(z) omits no complex number of large absolute value.
{ z : Izi
}
z
:
5.6.6 Exercise. The Maximum Modulus Theorem (5.3.36) may be re formulated for as follows. If Q is a region in I E H ( Q ) and for each a in 800 ( Q ) and some sup M, for all in Q, M, inf :::; M.
Coo
C,
N(a) EN(a) z EN(a)nn
I /(z) l :::;
z
I /(z) 1
[Hint: If Q is bounded 5.3.36 applies. If Q is not bounded and > M, for some positive E, each m in N, and some in
sup n Q,
Zm
I /(z) 1
I Zml > m and II (zm ) 1 > M + E.] Coo )
5.6.7 Exercise. ( Minimum Modulus Theorem for If I E H ( Q ) and o tJ- I(Q), for a in Q, Equality obtains iff I is a constant 2': inf
I /(a) 1
function.
I/( z) l . z En
[Hint: The function g �f
-7 is in H (Q ) ; 5.3.36 applies.]
n +1 sn+l n + lC 2 contains the set of all n + I-tuples (Zl, . . . , znn++dl such that L I Zkl > o. There is a relation among the elements of s : k= l {(Zl, . . . , zn + d (W I , . . . , wn + d } { 3 >. 3 /-t{ { 1 >' 1 + I IL I > O} 1\ { >' Zk + ILW k = 0, 1 :::; k :::; n + I}}} . The set
rv
{:}
rv
5.6.8 Exercise. The relation
rv
is an equivalence.
Chapter 5. Locally Holomorphic Functions
254
Sn+ l
n
5.6.9 DEFINITION. p ( C) �f / "' . Of particular interest for the context Coo and the map e defined earlier is p i (C) , the complex projective line. 5.6.10 Exercise. The map
pi
is a bijection between
otherwise (C) and Coo .
5.6.11 Exercise. If a E P(f), for some N(a) and some N(oo),
f[N(a) \ {a}] v . 5.3.39.
::J
[N(oo) \ {oo}] ,
(1[)),
5.6.12 Exercise. If f E H then 0 is a removable singularity of f iff 1R(f) or CS(f) is bounded near o. [Hint: 5.4.3 and 5.6.11 apply.] 5 . 7. Contour Integration
The Residue Theorem finds application not only in the theory of C-valued functions defined in some n, but also in the evaluation of definite integrals and in the summation of certain series.
1 1 2k dx, k E N, is, for positive R, reI. f 0 < t -< -1 Re27rit 2 /'(t) ' 1 1 -R + ( t - 2) 4R if -2 -< t -< 1 for which is the union of a semicircle and an interval [- R, R] . Di 1 rect calculation shows that if R 1 , on + z2k I 2': R2 k - 1 . Hence, 1 R , whence lim r 1 k dz = O. 1 dz l :::; R2: if R is large, I r k 2 -1 1 J l + z2 J l+z in n + �f { z : "-5(z) O } , Thus, if is the sum of the residues of l + z2k 1 dx 27riL The set P+ of poles of l +1z2k in n+ is then 1 IR I + x 2 k 5.7.1 Example. The integral Jr IR + x lated to the curve (contour)
=
/'
{
*
>
rR rR ,
R -too r R
rR
L
>
=
Section 5.7. Contour Integration
z
255
z) > 0, 1 + z2k = 0 } .
1 The residues of 1+ P + can be calculated via the formulre in 5.4.24. {
'S(
:
z2k
5.7.2 Exercise. ( Jordan's inequality ) If 0 :::; t :::;
at the points of
� , then � t :::; sin t.
[Hint: The geometry of the situation provides the clearest basis for the argument.]
1
00 -•
sln x 5.7.3 Example. The integral dx can be treated by an integration x o e iz of - over the curve "( defined in terms of the positive parameters E ( small ) and R ( large ) :
z
4
E + ( R - E)t Re4 7ri ( t - i )
4(
( �)
-R + - E + R) t Ee i ( ( � - t )4 7r + 7r )
-
. < t < 1 1f 0 1 1 and f E H [ ( , r r ] there r) , � f. r) , of polynomial functions such that on is a sequence b) If Q lr \ and f(z) �, then f E H (Q) but no sequence z of polynomial functions converges to f uniformly on compact subsets of Q. [Hint: a) Fejer's Theorem (3.7.7e) ) and the Maximum Modulus Theorem (5.3.36) apply.]
0 {Pn} nEN �f D(O, {O}
{ Da
�f
n C (D(a, C] }, D( a, Pn
5.9.11 Exercise. In 5.3.43 a satisfactory value for p is:
5.9.12 Exercise. A net of functions is locally uniformly convergent resp. locally uniformly Cauchy resp. locally uniformly bounded on an open set U iff U is the union of open sets on each of which the net is uniformly convergent resp. uniformly Cauchy resp. uniformly bounded.
Section 5.9. Miscellaneous Exercises
267
5.9. 13 Exercise. (Montel) A subset F of H(Q) is normal iff F is locally bounded, i.e., iff for each compact subset K of Q and some in JR.,
MK
xEK JEF
sup I f(x) 1 :s;
MK .
[Hint: 1.7.21 applies.] 5.9. 14 Exercise. If: a) l'(t) = e 2 7ri t , O :S; t :s; 1; b) for some ¢ in C 1 (D(0, 1), q , ¢[D(O, 1) ]
�f QC and for l' as in a) ,
and ¢ 0 l' is a Jordan curve such that aQ = (¢ 0 1')* ; c) f and g in Q 3 (x, y) r-+ [J(x, y), g(x, y) ] E JR.2 are continuously differ entiable Stokes's Theorem in the form (5.8.6) is valid.
5.9. 15 Exercise. If {u, v } c C OO (1U, JR.) , det
[��::�n
>
0, and
Q cle=f { [u(x, y) , v(x, y) ] : -0.5 < x, y < 0.5 } ,
r W ( l ) = r dw ( l ) . ln lo(n) [Hint: Both 5.9. 15 and the discussion in 5.8.5 apply.] [ 5.9.16 Remark. The result 5.9.16 provides extended circum stances where Stokes's Theorem applies. An argument based on patching together squares like Q �f { (x, y ) : -0.5 < x, y < 0.5 } and their images under maps like
then Q is relatively compact and for any I-form w ( l ) ,
T : Q 3 (x, y) r-+ [u(x , y), v(x, y)] E Q C JR.2 above leads to very general forms of Stokes's Theorem [Lan, Spi] . If ),2 ( Q ) = 00,
In dw ( l ) need not make sense.
For example, if
I }, then ), 2 (Q) = 00 (and Q is not relatively com pact). For the I-form w ( l ) �f -y dx + x dy, dw ( l ) = 2 dx 1\ dy and
Q �f { z : I z l
>
r
lo(n)
w ( l ) = 27r
-j. lnr dw ( l ) = 00.]
Chapter 5. Locally Holomorphic Functions
268
( � ) ( -� ) : such that g ( � ) = g ( -� )
5.9. 17 Exercise. a) There is a function I in H (D(O, It) and such that =I = 2 ' b) There is no function g in H (D(O, 1) 0 ) and I =
:3 '
5.9.18 Exercise. If I E H (fl) and 5 is a constant or Z(f) n 5 ° -j. 0.
f
� {z :
1
I I(z) l :S; I } c fl, either I
�
-
I Z) dz exists for a (z b) (z I z l=r all large r. b) If I I I is also bounded, then for each a, J'(a) = ° (a second 5.9. 19 Exercise. If I is entire, then: a)
proof of Liouville's Theorem.)
5.9.20 Exercise.
If
00
L an zn converges in 1U and:
n =Q
the recursion formula for the coefficients in
a) ao
00
-j.
0, what is
L bn zn that represents the
n=O
17 b) a l -j. 0, what is the recursion formula for the coefficients in L cn z n that represents the n= O ( inverse) function h such that near 0, h 0 I(z) z7 (reciprocal) function g such that near 0, g . I(z) 00
==
==
5.9.21 Exercise. If I(x, y) f u(x, y) + iv(x, y) and both . for some nonnegatIve . R, • ble at z clef · U" = x + zy, dluerentm I(z + h I(z) : h -j. ° = 8[ D( a , R) ] .
{
�
�-
u
and v are
r
o = pe ' , p > 0, ¢ fixed, the calculation [Hint: For z clef = re' and h clef I(z + h) I(z) as + ° applies.] p of h
-
°0
(10,z)
5.9.22 Exercise. HI � Z. [Hint: For k in Z and "Ik : [0 , 1 ] 3 t r-+ ( 0.S ) e 2 k".it , "Ik = k::;i . If is a rectifiable loop and C 10 then for some k in Z, Ind 1J (O) k. then I is representable by a Laurent series.] If I E H
(10),
5.9.23 Exercise. a) If "Il (t) f -1 + e 2 7ri ( l - t ) , "I2 ( t )
and r
f
�
� ::;i - ,.y;, then r s: U
1]
1]*
=
f
f
� 1 + e27rit , fl � C \ ({-I} U {I}),
1] 1 and 1]2 , 1 clef= 1]1 . "11 . 1]1- 1 and u2 clef= 1]2 . "12 . 1]2- 1 �n
O. b) For some curves s:
Section 5.9. Miscellaneous Exercises
269
are loops and 8 1 (0) = 82 ( 0 ). c) {} �f {8 d {J2 } {8 � 1 } {82 1 } -j. 1 but h ({}) = 0 (in Hd Q, Z)] , v. 5.5.28. 5.9.24 Exercise. The conclusion in 5.3.62 can fail if the condition on Q is replaced by on a set S such that S e Q and Q ::) S· -j. (/) as in 5.3.52. . z n + cle=f G ( z ) , then G E H (C \ 1I') , 5.9.25 Exercise. If z -j. and hm
1
I l
and
G( z ) =
{ 1-I
n---+ =
1 zn - 1
--
if I z l < if I z l >
1. 1
ex:> 1 1, then the Dirichlet series '" � nz converges n= 1 and defines a function f holomorphic in Q �f { z : �(z) > 1 } . 5.9.26 Exercise. If �(z)
>
-
[ 5.9.27 Remark. Riemann's zeta function ( is defined and holo morphic in C \ {I} and ( I n = f. Furthermore, P(() = {I} and Ord «(l) = It is known that Z(() C { a + it a .:s: I } and that Z ( ( ) is symmetric with respect to both a + it a =
1.
{ : O .:s: Riemann conjectured that Z (( ) C { a + it
:
:
�} a = � }.
and JR.. As of this writing, his conjecture remains unresolved, despite the efforts of some of the greatest analysts since Riemann's time. Rie mann's zeta function is of central importance in number theory, particularly in the study of the distribution of prime numbers, i.e., the cardinality 7r ( x ) of the set of prime natural numbers n such that n .:s: x. In J. Hadamard and C.-J. de la Vallee Poussin, using properties of (, independently proved . THE PRIME NUMBER THEOREM. x---+ hm (7r (xx ) = = ln x
1896,
) 1.
-
1948,
In P. Erdos and A. Selberg [Sha] proved the Prime Number Theorem without recourse to the methods of complex analysis.]
6
Harmonic Functions
6.1. Basic Properties
The subject of harmonic functions appears in 5.3.57. The conclusion to be drawn from 5.3.59 is that for some regions Q and some u in Ha lR (Q) , there is in Ha lR (Q) no function v that serves as a harmonic conjugate to u throughout the region Q. The next results explore other possibilities.
6.1.1 Exercise. a) If I �f u + iv E H(Q) , then
I E Ha(Q) and {u, v} C Ha lR (Q). b ) There are in Ha lR( Q ) functions u and v such that u + iv tt H(Q) . 6.1.2 LEMMA. a) IF u E Ha(Q) AND D(a, rr C Q, THEN THERE IS A v SUCH THAT I u + iv E H [D(a, rr] ' i.e., CONFINED TO D(a, rr , v IS A HARMONIC CONJUGATE OF u. b) IF, CONFINED TO D(a, rr, V I AND v2 ARE HARMONIC CONJUGATES OF u, THEN FOR SOME REAL CONSTANT C, V I - v2 = C. PROOF. a) If u E HaIR (D(a, rr) and a �f a + i(3, direct calculation, in view of the existence of Uxx and U yy and the validity of t:m = shows that x U y ( s , (3) ds is such the function v : D(a, rr 3 x + iy r-+ ux (x, t) dt -
�
iY
I
0,
that T : D(a, rr 3 (x, y) r-+ [u(x, y) , v(x, y)l has a derivative and further more that Ux = Vy and u y = -vx . Thus I u + iv E H [D(a, rrl . If u = �(u) + i<S(u) �f p + i q E Ha [D(a, rt] ' then
�
{p, q} c HaIR [D(a, rn . The previous argument implies that for some � and 1] in HaIR (D(a, rr), {p + i�, q + i1]} C H [D(a, rt] ' whence p + iq + (i� - 1]) �f u + iv E H [D(a, rtl . b) If Ij �f u + iVj E H [D(a, rtl , j = 1, 2, then i (II - h ) is JR.-valued in ( D(a, rt) and the Cauchy-Riemann equations imply that for some real constant c, i (II - h) = -c, i.e., V I - V2 = C. 0 270
Section 6.1. Basic Properties
271
6.1.3 COROLLARY. IF u E Ha lR (Q) , a �f a + i(3, AND D ( a , rr c Q: a) THERE IS A SEQUENCE {PTnn } :, n=o OF CONSTANTS SUCH THAT IN D( a, r r , u ( x, y ) =
00 , 00
L
PTnn (X - a ) Tn (y - (3) n ; b) THERE ARE SEQUENCES {Cn } nE Z '
Tn,n=O { an } nE Z+ ' AND {bn } nE Z + SUCH THAT IF 0 :::; R < r, THEN u (a + Re i O ) =
00
00
L
cn Rn e in O = L an Rn cos n O + bn Rn sin nO. n== - (X) n=O o
PROOF. The argument in the Hint following 5.3.25 applies. 6.1.4 Exercise. If r > 0 and u E HaIR [D ( a , rr ] ' for some v, v + iu E H [D ( a , rr ] .
6.1.5 LEMMA. IF u E Ha(Q) , D ( a , rr c Q, AND 0 :::; R < r, THEN
1 1 2". u (a + ReiO ) dO,
27r 0
(6.1.6)
u( a ) = -
i.e., u ENJOYS THE Mean Value Property MVP AT EACH POINT OF Q. [ 6.1. 7 Remark. Customarily the symbol MVP(Q) is reserved for the set of functions continuous in Q and enjoying the Mean Value Property at each point of Q; there is a corresponding meaning for MVPIR(Q). Thus 6.1.5 may be viewed as the assertion: Ha(Q) c MVP(Q) . The reversed inclusion is the burden of 6.2.16 below.] PROOF. For the harmonic conjugate v that serves in D ( a , rr ,
f �f u + iv E H [D ( a , rr] . o
Cauchy's formula applies.
6.1.8 THEOREM. (Maximum Principle) IF u E MVPIR (Q) , a E Q, AND
(6.1.9)
{
u( a ) 2': sup u ( x, y ) : ( x, y ) E Q } ,
THEN u IS A CONSTANT FUNCTION. PROOF. The MVP asserts that if D(a, R) C Q, the value of u at the center of D( a, R ) is the average of its values on 8[D( a, R)] . Hence, if m R resp. MR are the minimum resp. maximum of u on 8[D ( a, R)] , then implies
(6.1.6)
Chapter 6. Harmonic Functions
272 m R :::;
(6.1.9)
u(a) :::; MR and implies that for all R as defined, u(a) = MR . :::; R2 and u (xo, Yo) < u(a ) , then
If Xo2 + Yo2 clef = s2
{ (x, y) : X2 + y2 = S 2 , U(X, y) < u(a) } is a nonempty open subset of 8[D(a, s) ] . Thus
1 1 2". u (a + seiO ) de < u(a),
27r 0
u(a) = -
a contradiction. Hence, if u is not constant, for any a in fl, u(a) cannot be a local maximum value of u: is denied. 0 6.1.10 Exercise. If u E HaIR [D( a, r] n C[D(O, ) C] , then is valid when R = [Hint: When R < 6.1.5 applies. Passage to the limit as R t is justified by the Dominated Convergence Theorem (2.1. 15) and 6.1.8.]
(6.1.9)
r.
r
r
,
(6.1.6)
r,
r
6.1.11 THEOREM. (Maximum Principle in Coo ) IF fl c C, u E M V P IR(fl) , AND
{
u(x, y) sup {(a, b) E 8oo ( fl ) } '* N a b inf [( , ) ] EN[ ( a , b ) ] ( x , y) E N [( a , b) ]nn
:::;
THEN u 0 OR u(fl) c ( - 00 0) PROOF. If the result is false there are two possibilities: a) for some (xo , Yo) in fl, u (xo , Yo) > 0; b) for some (xo , Yo) in fl, u (xo, Yo) = 0 while u(fl) C (-00, 0] . If a) is true, for some positive E, =
,
o} ,
.
If K is unbounded, then 00 E 8oo (fl) and for a sequence {(xn , Yn ) } nEN in K, (xn , Yn ) -+ 00 as n -+ 00. Hence lim u(x, y) 2': E > 0, a contradiction of ( x , y) ---+ oo the hypothesis. Hence K is bounded and since u is continuous, K is closed: K is compact. Thus, for some (p, q ) in K, u(p, q) = max { u(x, y) : (x, y) E K } 2': E. If (x, y) E fl \ K, then u(x, y) < E , whence
u(p, q) = max { u(x, y) : (x, y) E fl } ,
and 6.1.8 implies u = u(p, q ) 2': E, a contradiction of the hypothesis.
in a Disc
Section 6.2. Functions Harmonic
If b) is true, 6.1.8 implies
u
273
u == o.
o
6.1.12 Exercise. If { , v} c MVPIR( fl) and for each point (a, b) in 8oo (fl) sup
inf
N [ ( a , b ) ] EN[ ( a , b) ] ( x , y ) E N [( a,b ) ] nn � on fl, either
u(x, y)
sup
inf
N [ ( a , b) ] EN[( a , b) ] ( x , y) EN [ ( a , b) ]nn
u < v or u == v.
[Hint: 6.1.11 applies to
u - v .]
3x
x
v( , y),
r-+ x
6.1.13 Example. a) Although f : C is in Ha lR (C) , f 2 is not + iy 2 harmonic. a) both g : C z e Z and g are harmonic. c ) The map h:C + i y y is in Ha lR (C) ; h 2 is not harmonic but f 2 - h 2 E Ha lR (C) .
3x
3 r-+
r-+
6.1.14 Exercise. If f E H (fl) and I f I E Ha lR (fl), then f is a constant function. [Hint: Since bo l f l = if f + iv, then Ux + Vx = U y + Vy = o. The Cauchy-Riemann equations imply ! , 0.]
�f u
0,
==
u
6.1.15 Exercise. If f E H (fl) and E Ha [J(fl)], then [Hint: If a E fl , for some positive r and positive s ,
u
0
f E Ha( fl ) .
D(f(a), s t C J [D(a, rn c f(fl), and 6.1.2 implies for some v, g g o f E H ( fl).]
�f u + iv E H {[D(f(a), st] }, i.e.,
6.2. Functions Harmonic in a Disc
00
00
1 = ( 1 + z) � '" z = 1 + 2 � '" z is in H (1U). 1U 3 z r-+ � 1-z Hence 'iR(f) E HaIR (1U). Customarily, polar coordinates are used to discuss 'iR(f) in 1U, and when z = re i O , 0 :::; r < 1, fJ E JR.,
The function f :
n
n
n= l
n=O
1 - r2 2 sin fJ 1 + re iO --,."...- -----;,f ( z ) = 1 reiO = 1 - 2r cos fJ r2 + i -.... 1 - 2r cos fJ + r2 ' + l re i O 1 - r2 'iR + i = Pr(fJ) . 1 - 2r cos fJ + r 2 1 - re O
{
The map
}
�f
1U 3 (r, fJ) r-+ Pr (fJ) is the Poisson kernel.
Chapter 6. Harmonic Functions
274
6.2. 1 Exercise.
00
n= - (X)
�f
0
6.2.2 LEMMA. IF :::; r < R AND Ca(R) { z : I z - a l = R }, FOR THE COMPLEX MEASURE SPACE (Ca(R), 5(3 [Ca(R)] , p,) ,
h : D(a, r t
3 re iO
1
r-+
Ca r r)
IS IN Ha [D(a, Rt] . PROOF. Since PI- (8 - t) R
=
�f
Pfi ( 8 - t) dp, (a + Re it )
{ �},
�f h(r, 8)
eit + PI- is in HaIR (D(O, Rt). Be. re ' e,t - R a + i(3, when z = re iO , h is a complex linear
�
R
cause p, = �(p,) + iSS(p,) combination of the real and imaginary parts of
cl f r27r
.
r27r
. Re it + z Re it z dO' (a + Re't ) + i J Re i + d(3 (a + Re't ) . I(z) � J Rei t z t z o o By virtue of 5.3.25, each integral above represents a function holomorphic in D(a, Rt · 0 [ 6.2.3 Remark. On Ca( R) arc- length may be used as a basis for a measure space [Ca(R), 5(3 [Ca(R)] , �] that is the analog of T in 4.5.2. The group '][' acts on Ca (R) according to the rule '][' x Ca (R) {e iO , z } r-+ a + e iO (z - a). _
_
3
With respect to the action of '][' on Ca(R), � is action-invariant : if E E 5 (3 [Ca(R)] and :::; 8 < 271", then
0
a + e io (E - a) E 5(3 [Ca (R)] , � [a + e i o (E - a)] = �(E) .
clef dp,
If p, « � and g = � ' by abuse of language, 6.2.2 says the convolution Pfi g is harmonic. *
When k defined on Ca(r) is such that
r27r
Jo Pfi (8 - t)k (a + re it ) dt
Section 6.2. Functions Harmonic in
a Disc
275
exists, the result is the Poisson transform of k and P(k) . It is a harmonic function defined in D ( a , r r . The role of the family {PI.. } R
0< _ 1'
(1 _ r ) 2 + 4r� . Thus 2 7r z
==
•
7r
if "2 < IfJl
M, 6.3.7 ap plies when u ( ) M and leads to a contradiction. Hence u l n :s: M. If r > 0 and b E D(a, r) c 0., for some sequence {u n } nEN in F(n, I), n---+lim= un (b) = Uf(b). The sequence { wn �f U l V . . , V U n } nEN is part of PROOF. If
==
z
F( n, I). Furthermore, the functions
n ( ) �f { wPn(w(z)n ) ( )
w
z
z
if z E D(a, r)O otherwise '
n
E N,
are in F(n, I), are harmonic in D(a, rt, and
Harnack's Theorem (6.2.24) implies that if z E D(a, rt,
exists and nition,
Wb E HaIR [D(a, rt] ' whence Wb (b) = Uf (b). Moreover, by defi-
If C E D( a, r t, F(n, I) contains a sequence
If
Yn cle=f U n V Un , E N, and Yn (Z) �f { YnP ((z)Yn) ( ) -
(6.3.17)
{un } nEN such that
n
z
if Z E D( a, r)O ' otherwise
n
E N,
Ye
n---+ = Yn
in the preceding paragraph, if Z E D( a, r t , then ( z ) �f lim ( z ) ex= Uf ( ). The analog of (6.3.17) ists and E HaIR [D(a, rtL whence
as
IS
Ye
Ye (c)
c
which is valid by virtue of the definition of the
Wb :s: Ye ,
Yn ' Hence
(6.3.18)
V �f Ub , T - Ue,T :s: 0 and so V(b ) = O. The Maximum Prin Since ciple implies V I D ( a , T) O == 0, i.e., = Uf ( ). Since is any point in D(a, rt, Uf E HaIR [D(a, rtl; since D(a, rt is any open subdisc of 0.,
Uf E HalR (n).
Wb(c)
c
c
0
Chapter 6. Harmonic Functions
290
6.3.19 Exercise. The function Uf (the Perron function associated with Q and f) is unique. Via 6.3.15 there can be constructed a function Uf harmonic in Q and, by abuse of language, majorized by f near r �f 8oo ( Q ). Since r is compact, the condition that I f I be bounded is satisfied if f is continuous. 6.3.20 Exercise. A Perron f-family F �f F(Q, f) is a Perron family, viz., a set F contained in SH(Q) and such that:
1\
{{v E F} {D(a, Rr c Q}} ::::} {Va , R E F} , {{V l , V2 } c F} ::::} { :3 (V 3 ) {{V3 E F} {V3 2: max {v l , v2 }}}} '
1\
(6.3.21 ) ( 6.3.22)
6.3.23 THEOREM. IF F IS A PERRON FAMILY, V cle=f sup v, AND V < 00, F THEN V E HalR(Q) .
(6.3.22)
PROOF. The condition permits the replacement of convergent se quences selected from F by monotonely increasing sequences in F. Hence Harnack's Theorem is applicable. An argument similar to that in the PROOF of 6.3.15 applies. 0 Absent topological conditions on Q, Q can fail to be a Dirichlet re gion, e.g., if Q = 10 � 1U \ {O}, v. 6.3.1l. When Q = 10 the following are noteworthy features: •
•
�f 8oo ( Q) = {0}l:J1I';
r
HaIR (Q) n C ( Qc , JR.) contains no function (3 such that (3(z) is
{ positive zero
�f z E r \ {O} (= 1I') . If z = 0
P
(Indeed, for such a (putative) (3, ((31 r ) �f v E HaIR ( 1U) n C (U , JR.) . Since (3 is positive on 1I' and v E MVPIR(1U) , v(O) = v(O) - (3(0) > O . If, for some a in 10, v(a) - (3 ( a ) �f E > 0, for z in 1Uc and
l
p(z) �f v(z) - (3(z) + E l z l 2 ,
p T = E and p( a ) > Hence for some b in 1U, p( b) 2: max p(z) . Consez El[JC quently Px (b) = p ( b) = 0 and Pxx (b) < 0, p ( b) < 0: t:J.p(b) < O. On the other hand, since v - (3 is harmonic in 10, t:J.p = 4E > 0, a contra diction. Thus v - (3110= 0 and zlim v(z) - (3(z) is both v(O) (> 0) and, --+o by virtue of continuity, 0, a contradiction: no (3 as described exists.) y
Eo
The preceding considerations motivate
yy
Section 6.3. Subharmonic Functions and Dirichlet's Problem
6.3.24 DEFINITION. FOR A REGION Q AND AN a IN r TION (3 IN HaIR (Q) n C (Qc , JR.) SUCH THAT
(3(z) is
positive { zero
291
�f 800 (Q) , A FUNC
IF Z E (r \ {a}) IF Z = a
IS A barrier at a. 6.3.25 Exercise. If Q is a Dirichlet region, there is a barrier at each point a of 800 (Q) .
�f { -�, �}
6.3.26 Exercise. If Q 1U \ there is no barrier at either 1 or 2 . (Hence Q is not a Dirichlet region. ) For 6.3.25 there is a converse derived from
-�
�f
6.3.27 THEOREM. IF Q IS A REGION, a E 800 (Q) r, AND THERE IS A BARRIER AT a, FOR EACH f IN C (r, JR.) AND THE ASSOCIATED UNIQUE PERRON FUNCTION UJ , lim UJ(z) = f(a) . zEn
PROOF. If E > 0, for some positive r, Z E D( a, r t n r implies
I f(z) - f(a) 1 < E.
�f
The hypothesis implies that (3 has a positive minimum m on the compact set r \ D(a, rt. If M Il f lloo the function :
u Q C 3 Z r-+ f(a) + E + is in HaIR (Q) , and
(3(z) m
[M - f(a)]
{
u(Z) -> f(a) + E > f(z) if z E [D(a, rt n r] M + E > f(z) if z E {[C \ D(a, rt] n r}. If v E F(Q, f), by virtue of the Maximum Principle in Coo , whence (UJ - u) I n :::; o. Therefore, inf
sup
N (a) EN(a) zEN (a)nn
UJ(z) :::; u(a) = f(a ) + E.
On the other hand, the function w
:
QC
3
(3(z) z r-+ f ( a) - E - - [M + f ( a ) ] m
Chapter 6. Harmonic Functions
292
is in HaIR(Q) , and
{
E < J(z) if z E [D(a, r) O n r] w(z) < J(a) -M E < J(z) if z E {[C \ D(a, r) O ] n r} . -
-
-
Hence the harmonic function w is in F(Q, J) and so (UJ sup inf UJ(z) 2': w(a) = J(a) E.
-
w) I n 2': 0, i.e.,
0
-
N(a) EN(a) zEN (a)nn
6.3.28 COROLLARY. IF THERE IS A BARRIER AT EACH a IN r, FOR EACH J IN C(r, JR.) , DIRICHLET'S PROBLEM HAS A UNIQUE SOLUTION. PROOF. The result 6.3.27 applies. [ 6.3.29 Note. Aside from useless tautologies, there seems to be no general necessary and sufficient condition for the existence of a barrier at a point a in 800 (Q) .
o
On the other hand, a sufficient condition for the existence of a barrier at a in 800 (Q) is the following. If Q, a E 800 (Q) , and the component of Coo \ Q that con tains a is not a itself, there is a barrier at a, v. 8.6.16.] The following observations provide some orientation about Dirichlet regions and non-Dirichlet regions. If Q is a simply connected proper subregion of C, then Q is a Dirichlet region. ( Owing to 8.1.8d ) , Coo \ Q consists of one com ponent and is not a single point. ) Although 800 (q = 00 ( a single point ) , nevertheless C is a Dirich let region. ( The (constant ) function u == J(oo) is a solution of Dirichlet's problem. If 0 < r < R :::; 00 , the region A( a; r, R) 0 , which is not simply con nected, is a Dirichlet region. The regions 10 ( = A(O; 0, It) and C \ {O} are not Dirichlet re gions, v. 6.3.11. 6.3.30 Exercise. If r > 0 , D(b, rt n Q = 0, and a E {[800 (Q)] n Cb( r) }, a is a barrier at a. then (3 : QC 3 z r-+ In In z
(�)
are:
-
I
-
; bI
Alternative definitions of the notion of a barrier at a
m
r
�f 800 (Q)
Section 6.3. Subharmonic Functions and Dirichlet's Problem
293
(3 N(a) n Q, (N(a) n Q)C , �f Zz E [(N(a) n Q)C \ {a}] . (3(Z) (3r {(3r } r>O Qn ( (3r(z) (3r(z) EQn ( , (3r(z) (3 E SH(Q); (3 E 8(Q) \ {a}, z=b 6.3.31 Exercise. If Q is a Dirichlet region, for each a in 800 (Q), there is, in the sense a) or b) a barrier at a. 6.3.32 Exercise. If, in the sense a), (3 is a barrier at a, then lim U (z) f(a). z---+ a J 6.3.33 Exercise. If, in the sense b), (3 is a barrier at a, then lim U (z) f(a). z---+ a J 6.3.34 Exercise. If, in the sense c), (3 is a barrier at a, then at a there is a barrier as defined in 6.3.24. [Hint When f (3, 6.3.15 applies followed by a change of sign.]
a) ( [Re, Ts] ) A barrier at a is a function defined in some open neigh borhood a ) continuous on subharmonic on and such that is { positive zero If = a such that: a) is su b) ([Con] ) A barrier at a is a family perharmonic in D a , rt and 0 :::; :::; 1; b) lim = 0; c) if z---+ a w Ca r ) then z---+ limw = l. c) ([AhS, B e] ) A barrier at a is a function such that: a) then lim(3(z) < o. b) z--+a lim (3(z) = 0; c) if b
N( ,
=
=
:
=
For some purposes the discussion of subharmonicity is carried out in the following more general context.
u u(Q) [-00, (0); u E un ..l-
6.3.35 DEFINITION. A FUNCTION IN JR.r1 IS subharmonic in the wide sense, i.e., U C b) usc ( FOR A IFF: a) SEQUENCE OF CONTINUOUS FUNCTIONS, U, v. 1.7.24) ; c) FOR EVERY a AND EVERY POSITIVE R SUCH THAT D( a, Rt C
E SHW(Q), {Un} nEN
1
1 271"
U ( a + Re iO ) dO, u(a) :::; 0 n
27r E
n
E N.
Q,
(6.3.36)
u(Q) c [-00, (0), u E SHW(Q) IFF Q, { {v E HaIR (KO ) n C(K, JR.)} 1\ { v l a( K) 2: U l a ( K) } } ::::} { v i K 2: u l K } ' (6.3.38)
6.3.37 THEOREM. IF U usc(Q) AND FOR EVERY COMPACT SUBSET K OF
Chapter 6. Harmonic Functions
294
i.e., IFF DOMINATION ON 8(K) OF U BY A HARMONIC FUNCTION v IMPLIES DOMINATION OF U ON K BY THE SAME v. PROOF. If U E USC(Q), U (Q C [ 00 , (0 ) , and U E SHW(Q) , the averaging argument used for 6.1.8, 6. 1.11, 6.1.12, and 6.3.8 yields, for each compact subset K of Q and each v in HaIR(Q), the inequality For the converse, if r > and D( a, r) 0 C Q, for some sequence of functions continuous on D( a, r) , ..l- Thus
)
0
P
-
(6.3.36) .
{un } nEN
Un u.
{ Vn �f (Un ) } nEN c HaIR [D(a, rtl ,
Thus U E SHW(Q) .
{un} nEN C SHW(Q), AND b ) { Z E Q } { lim Un (z) � C < oo} , n---+=
o
6.3.39 THEOREM. ( Hartogs ) IF: a ) ::::}
FOR EACH COMPACT SUBSET K OF Q AND EACH POSITIVE E THERE IS AN no(K, E ) SUCH THAT ON K
(6.3.40) (For the significance of a ) , v. 6.3.42) . PROOF. For some M and all n, � M on K since otherwise, K contains a sequence such that for some Z= in K and some sequence > n. Then, because each is usc, n , lim Z = Z , UTn n
{zn} mn 2': n---+=nENn =
Un (zn) n---+lim= Un (z= ) = 00 (> C),
Un
{mn} nEN '
a contradiction. Hence, for the purposes of the argument , the assumption
Un I K < 0 is admissible. Since Un is usc, K c U cle=f { z : u n (z) < o } E 0(((:) , whence u nl u < o.
Section 6.3. Subharmonic Functions and Dirichlet's Problem
295
Because K is compact, 1.2.36 implies that for some fixed positive r,
U D(z, 2rt C U. If Z is fixed in K and un (z) > -00, since Un E SHW(Q), zEK
0,
Since u n l u < if v E U and Iz - v i < rS < r, then replacing r by r + rS in the last inequality leads to 7r(r + rS) 2 un (v) :::;
r
JD(v,r+W
un (w) dA 2 (W)
(6.3.41 )
:::; rD(v,r O un(w) dA 2 (W) . J
)
(6.3.40)
On the other hand, implies that if E is a measurable subset of D(z, 2rt, then lim r un (w) dA 2 :::; r lim u n (w) dAs :::; A 2 (E) Thus, n� = JE h n� = for some if > then u n (w) dA 2 < + E) . A 2 (E). then Consequently, if rS is sufficiently small (and positive) and >
Ie
nz, n nz ,
(C
(6.3.41) implies that if n > nz , then un (z) < C + E. for some finite set Z 1 , . · · , zm , K C U D (zm , rSt. If k= 1 nO ( K, E ) clef= 1 �k�rn max nZk
and
the desired conclusion follows. 6.3.42
Examp 1e.
[ = If K clef
.
C
n nz ,
Since r is z-free,
o
("\
0, 1] 2 , H clef= C,
if x > O , if x :::; °
1 n1 ' if x 2': n 1' f
X
0, w and I z then
r
a l 2: r,
E D(a, rt ,
Rw ( z )
= a - a) = ( --aa ) _ a a 1 z - - (w
Ra (z)
1
1-
z-
w
=� �
n=O
(w
(z
-
)
)n . n+l
( 7 .1.1) 301
Chapter 7. Meromorphic and Entire Functions
302
Convergence of the right member of (7.1.1) is uniform on the closed set 2: What follows exploits these elementary conclusions to reach a consid erable generalization. The global version of Cauchy's integral formula (5.4.14) may be viewed as a special case of the next result, which is the key to the derivation of Runge's Theorem.
{ z : I z - a l r }.
7.1.2 THEOREM. IF (O(C) \ 0)
3
U ::J K E K(C) , THERE ARE RECTIFI
N bn h sup I /(w) l . c)
wEK
PROOF. The pattern of proof is: c)
::::}
b); a)
::::}
b)
::::}
a) ; b)
::::}
c).
c) ::::} b): If b) is false. for some component e of U \ K , e c n U is com pact. If p E Be, some sequence {Pn }nEN contained in U converges to p. Since e is open, p tJ- e, whence p E e c n U c U, P E U \ e K: Be c K. The Maximum Modulus Theorem 5.3.36 implies sup I /(z) 1 :s; sup I /(z) l , a Cc K denial of c) .Hence -,b) ::::} -,c) (7. 1 . 17) =
and c) ::::}
::::}
b).
b): If b) is false and I E H(K), for some sequence {fn }nEN in H (U), In � I on K. Furthermore (7. 1 . 17) implies that for all n and m in N,
a)
Chapter 7. Meromorphic and Entire Functions
308
sup I /n (z ) - Irn (z ) 1 � sup I /n (w) - Irn (w) l · Thus {fn } nEN converges uniwEK formly on Ge to some F. Hence F is continuous on G e , F = 1 on K , which contains BG (v. preceding argument for c) ::::} b)) , and F E H(G). If w E G, there are an open V containing K and an N (w) such that V n N ( w) = 0. 1 __ . Then A particular choice of 1 is given by the formula I(z) z-w 1 E H(K) , (z - w)F(z) = 1 on an open set W containing K, and on the nonempty open set G n W. The Identity Theorem implies (z - w)F(z) = 1 on G. When z = w, there emerges the contradiction, 0 = 1 : a) ::::} b) .
z ECC
�f
b) ::::} a): If a) is false, I, viewed as an element of G(K, q , is not in the closure of H (U) viewed as a subset of G(K, q . The Hahn-Banach Theorem (3.3.8) implies there is a measure space (K, S ,B , p,) such that for every g in H(U), g dp, = 0, while 1 dp, = 1 . The following argument shows that K K the last equation is false. 1 The map ¢ : C \ K 3 w r-+ r __ dp,(z) is in H(C \ K) (v. 5.3.25). z w iK If w E C \ U, since z r-+ l k is in H(U), the Hahn-Banach Theorem (z - w ) + implies here that
i
i
1
k ¢ ( l (W ) = k!
1
r
k dp,(z) = 0, iK ( z - W ) + 1
00
1
k
E Z+ .
(7. 1. 18)
zn
-1 '
If I w l > sup I z l , then -- = - '" n + and the series converges uniz-W K n=O w formly for z in K. Since z n dp,(z) = 0, n E N+ , ¢( z ) = 0 if z is in the (unique) unbounded component of C \ K. If V is a bounded (open) component of C \ K and V n (C \ U) = 0, then V C U and the boundedness of V implies Ve n U is compact, a contra diction of b): V n (C \ U) -j. 0. If w E V n (C \ U), (7. 1 . 18) implies that for some neighborhood W of w, ¢ I w= 0, whence ¢ I v = o. In sum, ¢ I IC\K = o. If N is a neighborhood of K, the compactness of K implies that N may be assumed to be the union of finitely many open squares Q� . of the form in 7.1.2. For some infinitely differentiable '1jJ defined on N, '1jJ I on K and '1jJ 1 1C\ N = 0, in particular, '1jJ = o. If 1 E H( N), al = o. According to the product rule for derivatives, aU · '1jJ) = (af) . '1jJ + I · a'1jJ = I · a'1jJ. Hence Pompeiu's formula (5.8.14) implies that if z E K, then
i
l
�
==
iJN
I(z) = l (z)'1jJ(z) =
1 27r l
-.
1 I(w) B'1jJ(w) dw 1\ dw. N
--
W-Z
-
Section 7.1. Approximations and Representations
309
Fubini's Theorem and the definition of ¢ imply
r
}K
J(z) dpJz) = -�
=
1
(1
)
8'1jJ(w) dw 1\ dW dp,(z) J(w) W-Z K } J(w)8'1jJ(w) . ¢(w) dw 1\ dw.
27r l
r
N
i
Since ¢ I IC\ K = 0 and 8'1jJ(w) = 0 on K , J dp, = 0, a contradiction. (A similar technique is employed in the PROOF of Runge's Theorem.) r,
r
b) ::::} c): If z E U \ K, for some positive D(z, ) c U \ K. If C is a component of U \ [K U D(z, )] either C or C U D(z, ) is a component of U \ K. Thus b) obtains for K U D(z , ) Since K n D(z , ) = 0, there exist disjoint open neighborhoods N(K) and N[D(z , ) ] . Hence there is in H {N(K) U N[D(z, r)] } a function J such that J I K = 0 = 1 - J I D ( z , T) ' Since b ) ::::} a) , there is in H (U) a g such that r
'
r
.
r
r
r
o
from which c) follows.
7.1.19 Exercise. If K is compact, every function in H (K) is uniformly approximable on K by polynomials iff C \ K is connected. [Hint: a) In the current context, for U in 7.1.16 , C may serve. b) A component of C \ K is relatively compact in C iff C \ K is not connected. c) If J is entire, J is uniformly approximable on each compact set by polynomials.] The similarity of the techniques used in the argument for Runge's Theorem and its variant leads, by abuse of language, to the conclusion: Runge-variant ::::} Runge. 7.1.20 Exercise. For the set K in the argument for 7.1.2, no component of U \ K is relatively compact in U. (The condition 7.1.16b) may be in terpreted roughly as saying that part of the boundary of each component of U \ K meets the boundary of U, v. 7.1.15b) .) Since the components of Coo \ K play a role in the previous discussions, the following result is of interest, and proves central in 8.1.8. 7.1.21 THEOREM. IF Q IS SIMPLY CONNECTED, THEN F CONNECTED.
�f Coo \ Q IS
Chapter 7. Meromorphic and Entire Functions
310
oo J
PROOF. As a closed subset of C , if F is not connected, there are two dis joint, closed, and nonempty sets and such that F = 00 is in one, say and thus is compact. Furthermore O(q 3 U C \ = 1 If f 1 in 7.1.2, one of the summands, e.g. , __ dw, in (7.1.3) is
K
J,
==
K
Jl:JK; �f J Ql:JK.
1
/'1
w-Z
J) K (Ql:JK) K Q Q.
K
\ = and not zero. On the other hand, U \ = (C \ \ = 1 According if Z E then __ , as a function of w , is holomorphic in
K,
to 5.3.14b),
1
w-Z
/'1
1 __ dw = W - Z
0
0, a contradiction.
7.1.22 Example. The complement (in q of a nonempty compact subset of C is not simply connected. The region C \ ( - 00 0] is simply connected. Although the complement (in q of the strip S : - 7r < ( ) < 7r is not a connected subset of C, S is simply connected. The presence of Coo rather than C is essential in the statement of 7.1.21. ,
�f { z
'S z
}
7.1.23 THEOREM. (Mittag-Leffler) FOR AN OPEN SUBSET U OF COO , IF S C U, S· n U = 0, AND FOR EACH IN S THERE IS A RATIONAL FUNCTION ( Z ) = FOR SOME f IN P(f) = S AND (Z P (f) = PRO O F. For as in 4.10.9, the sets
a
ra clef LnN=(a1) cn (a)a) n ' ra · {Kn } nEN _
a
M(Q),
Pn ( z ) � aEL ra ( z ) is a rational function holomorphic in an open set containing Kn - I . Since Sn is finite, Coo \ Sn has only one compo nent Cn and 00 E Cn . Thus Runge's Theorem (7.1.11) applies and yields a rational function Pn for which 00 is the only pole: Pn is a polynomial. 1 on Kn - I , Furthermore, the Pn may be chosen so that I Pn (z) - Pn (z) 1 < and f � P I + L�= 2 ( Pn - Pn ) meets the requirements. 0 are finite, and
Sn
2"
7.1.24 THEOREM. (Weierstrafi) IF
U E O(q , S c U, AND S· n U = 0 , FOR SOME F IN H(U), S c Z(F) . PROOF. If the set S = ° resp. S =
{anh � n� N < oo' then F
==
1
resp.
Section 7.1. Approximations and Representations
311
5 {an } nEN 5 {anh S nS N < oo , U Ie z C, I (z) �f !c(z) , I C U U. U 0 tJ- 5. 0 5 fl �f U \ 5 a fl, Pn (z) �f z 1_ a n D(z, rt fl, a z fl,
requires meets the requirements. Thus only the possibility = motivates the attention. The paradigm for F when = argument below. The components of are pairwise disjoint regions. If is the solution for the component of and for in then is a solution for Thus , for ease of presentation, it is assumed that itself is a region and The situation for which E is dealt with in 7.1.26c) . The region is polygonally connected (1.7.11). In 7.1.23, when and is fixed in for a nonempty contained in and a polygon 7r; connecting to and contained in the function
-
__
In,7rz D(z, rt 3 z' r-+ exp 7rz + [z ,z 1 Pn (Z) dZ) IS H [D(z, r)O]. If 1]; is a polygon like 7rz , then 7T"z 1]z Pr> (Z) dz is in 27riZ, whence In, 7rz (z') is independent of the choice of 7rz : In, 7rz �f In. ' If Fn �f exp ( fn ), then ( z - an ) = 0, and direct calculation shows Fn ( z) z - an . For appropriate polynomials {Pn } l < < , the series that Fn (z) a - an 00 PI + nL ( Pn - Pn ) � I E M(fl). Furthermore F �f exp(f) is well-defined =2 in fl, F E H(fl), and
(1 1
;
m
= ___
I
1 -1
_ 11.
CXJ
z - a l II { z - an exp - Pn (W) dw] } . (7.1.25) (z) = -a - a l n2': 2 a - an The right member of (7.1.25) is an infinite product. Owing to the continuity of exp, the infinite product converges uniformly on compact subsets of fl and as z -+ an F (z) -+ O. Hence the set 5 consists of removable singulari ties of F and if F (an ) �f 0, n E N, then F E H(U) and 5 C Z(F). 0 The discussion of infinite products is given in Section 7.2 where the development implies Z(f) 5, v. 7.2.11. [ 7.1.26 Note. a) If I is required to have a zero of multiplicity f.-ln at an , the sequence {an} nEN may be modified so that each an appears f.-ln times. F
.
__
=
[1
'Y
312
Chapter 7. Meromorphic and Entire Functions
Z(f) �f {an} nEN ' then n, I z l < l anl , then 1 1 zk '"'" - . = = Pn = � z - an an ( 1 - -z ) k=O a�+l an Kn )k The approximating polynomial Pn is, for some Kn , - L ( : k=O ) n and the convergence inducing factor exp (-1 Pn (W) dw takes Kn Kn zk + l k ) . The exponent L zk + l k the form exp ( L k=O (k + l)an+ l k=O (k + l)an+ l f
f(O) i- 0, a 0,
b) If is entire, = and S· 0 and -+ 00. Hence, for each if
l anl
=
(Xl
( z - an )
is a partial sum of the familiar pOwer series representation of a de1 termination of - In _ . The Mittag-Leffler Theorem com bined with the argument above yields the _ _
WeierstrajJ product rep
resentation for the entire function f: f(z) = II n� l
{ (I - :n ) (tk=O (k zkl)a+l nk+l ) } .
f(O) 0,
exp
(7.1 .27)
+
k
c) If = for some in N, the product representation is pre ceded by a factor
zk .
Since the right member of (7. 1.27) converges uniformly on compact subsets of U and since the function exp is continuous, the validity of (7. 1.27) is automatic. Its derivation is independent of the theory of infinite products.]
[ f natural boundary f.
7.1.28 Exercise. If Z ( W = au, there is no function F such that: a) F is holomorphic in an open set V that properly contains U; b) F l u = Thus au is a for For such an F, if Z ( Ft = 0, then U = C . If Z ( Ft 0 5.3.52 applies.]
[Hint:
f.
i-
f
7.1.29 Exercise. If U E 0(((:) , then H (U) contains an for which au is natural boundary. In U there is a sequence S such that S· = au. In H(U) there is an such that = S.]
a
[Hint:
f
�f {zn} nE N Z(f)
Section 7.2. Infinite Products
313
7.1.30 Example. If a sequence of holomorphic functions converges every where, need the convergence be uniform on every compact set? In [Dav] the following construction uses Runge's Theorem to produce a sequence of polynomials such that
{Pn } nEN
if Z = 0 (7. 1.31) otherwise · Although the sequence converges uniformly on every compact set not con taining the sequence fails to converge uniformly on every compact set properly containing For in N, if �f > while < + lim
n --+ oo
{O}, n
Pn (Z) = { 01
{O}. Un { a ib : a - �, I bl clef
clef
�} Fn l:.J [; , n] ,
1 Fn = D (O , n ) \ Un and Kn = each Kn is compact, Kn C K� + l ' and U Kn = C \ {O}. Furthermore, for nEN each n, C \ Kn is connected and there are disjoint Open neighborhoods Vn of Kn and Wn of o. Hence there is a function gn holomorphic in Vn l:.JWn that
and such
if z E Wn if Z E V;, . Polynomial Runge (7. 1.14) implies there is a polynomial 1 < . Hence (7. 1 .31) is valid.
Kn , I gn - Pnl -n
Pn such that on
7.2. Infinite Pro ducts
The Weierstrafi product representation (7. 1 . 27) leads naturally to a discus sion of
infinite products.
7.2.1 DEFINITION. FOR A SEQUENCE f
00
{an} nEN OF COMPLEX NUMBERS,
infinite product P � II ( 1 + an) EXISTS IFF a) n= l N · PN 11· m II (1 + an ) = 11m N --+ 00 n N--+ oo =l EXISTS, IN WHICH CASE P = 11· moo P N ; b) FOR SOME no, N--+
THE
def
def
N
lim II ( 1 + N--+ oo
n=:no
an )
Chapter 7. Meromorphic and Entire Functions
314
EXISTS AND IS NOT ZERO. 7.2.2 Remark. The condition b) has the following motivations. A product of finitely many factors is zero iff at least one factor is zero, whereas, e.g., if each factor 1 + is nonzero
[
•
an -� an N 1 and yet lim II (1 + an ) 0. lim N--+= 2N N--+= n= 1 An infinite series converges iff every subseries arising from the ==
=
=
•
deletion of finitely many terms converges. The validity of the analogous statement for infinite products is assured by b). If
if n 1 an {-I otherwise ' 1 N N then lim II (1 + an ) = ° but lim II (1 + an ) does not N--+= n= 1 N--+= n=2 = exist. The condition b) eliminates II (1 + an ) from considn= 1 eration as an infinite product. =
=
As the developments below reveal, the simpler definition requiring
N
only that lim II (1 +
N--+= rL= l
an ) exist suffices in the context of repre
senting entire functions as infinite products.]
= 7.2.3 Example. If the series L bn converges and 1 + an exp (bn ), then n= 1 = lim bn 0, lim an 0, and the infinite product II (1 + an ) converges to n= 1 =
rL---.--t cx)
=
=
7.2.4 Exercise. If
{anL :SnSN
{an} nEN C eX ; c) L l an (x) 1 CONn= 1 VERGES UNIFORMLY ON X; d ) sup l an(x ) 1 < 00 , THE INFINITE PRODUCT 00 II [1 + an (x)] CONVERGES UNIFORMLY ON X AND DEFINES A FUNCTION n= 1 X f IN e FOR ANY PERMUTATION 7r N 3 n 7r(n) E N, 00 f(x) = II [1 + a7r(n) (x)] . n= 1 FURTHERMORE, f(b) = 0 IFF FOR SOME no , 1 + ano (b) = O. [ 7.2.6 Remark. The hypotheses c ) and d ) are independent. For )
7.2.5 THEOREM. IF: a X IS A SET; b) n E/'! xEX
f-t
:
example, if X = [0, 1] and: •
x
if E (0, 1] otherwise
•
x and an (x) = n n 2: 2, c ) holds and d ) does not . 2 If an (x) 1 , n E N, d ) holds and c ) does not. ] ==
'
PROOF. For some M, sup n E/'!
xEX
l an (x) 1 :::; M < 00 .
If K E N, there is an N
depending on K and such that (1, 2, . . . , K) C [7r( I ) , 7r(2) , . . . , 7r(N)] . If 1 - > > 0 there is a Ko such that sup < If 2 ex:>
E
L l an (x) 1 E. xEX n=K o K N K 2: Ko , PK �f II [1 + an (x)] , and nN �f II [ 1 + a 7r( n ) (x)] , n= 1 n= 1
owing to 7.2.4,
InN - P KI .:::; IPKI
( eC
-
1)
n :::; I P KI L E < 2 1 pKI E. n= 1 00
(7.2.7)
Chapter 7. Meromorphic and Entire Functions
316
Moreover, 7.2.4 implies that for some P and all K, I P K (X) I :::; P < 00 . If is the identity permutation, ( 7 . 2 . 7 ) implies that for some f, P K � f. Furthermore, if K > Ko , then
7r
I P K - P Ko I :::; 2 1 p Ko I E, I P K I 2': (1 - 2 E ) I P Ko I , {x E X} '* { I f(x) 1 2': (1 - 2 E ) I p Ko (x) I } .
Hence {J(x) = o } {} {p Ko (x) = a}. Finally (7.2.7) implies that for each x, lim n (X) = lim P N (X) . 0 N--+= N N--+=
7.2.8 COROLLARY. IF 0 :::; an < 1, n E N, THEN
N
PROOF. If P N �f II (1 - an ) , for some p , PN ..l- p. If n= 1
=
=
L an < 00 ,
7.2.5
n= 1
implies P > O. On the other hand, if L an = 00 , for each N, n= 1
N
whence lim II (1 - a n ) = o. N--+=
o
n=1
[ 7.2.9 Remark. The last sentence provides another motivation for b) in 7.2.1.] If z E 1U the series
ex:>
n
- L -=-n converges, say to l(z) . If g �f exp(l), then
( )
n= 1
g(Z) ' g'(z) (1 - z)' . , whence = o. Smce l(O) = 0, g(z) = 1 - z, I.e., 1-z 1-z g (z) (1 - z) exp[-l(z)] = 1 and so l(z) is a determination of In(l - z). It follows that for z fixed in 1U and K in N, for some N(K, z) in N, .
--
(
� 1 - (1 - z) exp t;Z �k =
(
)
N(K, )
II (1 - z) exp t;Z znn =
According to 7.2.3,
N (K' )
:::;
)
=
� 2 - K < 2. converges.
Section 7.2. Infinite Products
317
{an } nEN C C and 0 < I an I :::; I an+1 1 t oo, for any {N(K, n, z)} KEN such that k ( ) K, n ,z ( Z ) N ( ) :kn II 1 - an exp L n= 1 k= 1 converges to some number J(z). If N(K, n, z) can be chosen to depend only on n, i.e. , if there is a sequence {Nn } nEN such that More generally, if fixed there is a sequence
Z
00
-
(
)
f-t
J : C 3 z J(z) is entire and 7.2.8 implies J(b) = 0 iff for some z The expression E N (Z) �f (1 - z) exp (� : , the product of 1 - z and the exponential function of the sum of the first N terms of the Maclau rin series for a particular determination of In ( 1 - z), is approximately (1 - z) l -1 z 1. The next result serveS to estimate the error of the ap converges, =
n, b an .
.
--
proximation.
=
)
-
I z l :::; 1
N Z+ ,
THEN AND E 7.2.10 LEMMA. IF PROOF. There are nonnegative numbers kE
Ck ,
1 1 - EN (Z) I :::; I z I N +1 . Z+ such that
En (O) = 1 and there is a sequence {ddkEN of nonnegative numbers such that E�(z) _ z N ( 1 + f dk Z k . If k= 1 -1[o ,z] E� (u ) du 1 (Z) E N _ g N ( Z) �f- Z N +1 - --'-'�z'-;N:-;-+-:-1-- , then g N (Z) = L e k z\ e k 2': 0, k E N. If I z l :::; 1, then I g N (Z) 1 :::; g N (I) 1. k= 1 o Furthermore,
=
00
)
=
Chapter 7. Meromorphic and Entire Functions
318
a sequence 0 l art l :::; l an+ l l t oo, then NNcontains n +l
7.2.11 Exercise. a) If
;, (0') = 1 - 1 12 =
o
a
7.3. Entire Functions
1
Since a nonconstant entire function cannot be bounded, a study of the behavior of max -M'\ R ; particularly for large R, is in order. Izl= R R. Nevanlinna [NevI] developed this subject in a very significant manner. Only the introductory aspects of the material are treated below.
I I(z) 1 �f
I),
1 an ao 0, M (R ; I)
0,
= n g(z) �f � (:! ) O! '
r = II(z ) l l , z , = r .:s: e1 z1a } . b) If I(z) �f L Cn z n , by abuse of notation when Cn = 0, n=O 7.4.5 Exercise. a) p (f) = inf =
{a : a -
2:
.....
0, lim
n ln n
n---+ CXJ _ n I en I . 1 c) If p (f) < 00, then T(f) = - lim n I cnl ep (I) n = p(f) = lim
1
.....
£ill. n •
7.4.6 Exercise. (The Open Mapping Theorem for meromorphic func tions) If 1 is meromorphic in a region Q, then ( Q ) is an open subset of C= .
I
=
7.4.7 Exercise.
nEN I cnl < 00.
If
then sup
[Hint: # (S n 1UC )
I(z) = L Cn z n and S(f) n 1Uc = S(f) n 1I' = P( f ), n=O
< 00.]
II(z) l l lI'=
7.4.8 Exercise. If 1 E [H (1U)] n [C (1UC , C)] and K, 1 is a ratio nal function. [Hint: The Schwarz Reflection Principle and the Cauchy-Riemann equations apply.] 7.4.9 Exercise. a) For the meromorphic function
T : C \ {-i} 3 z f-t
z-z z + z. ,
--
T (n + ) = 1U. b) Is there an entire function 1 such that 1 (n + ) = 1U?
8
Conformal Mapping
8.1. Riemann ' s Mapping Theorem
In each of 5.5.8-5.5.11, 5.5.17-5.5.19, 7.1.21, and 7.1.22 a simply con nected region Q plays a central role. Combined with 8.1.1 below, the contents of the cited results pro vide a useful edifice of logically equivalent characterizations (v. 8.1.8) of simply connected regions in C. 8.1.1 THEOREM. (Riemann) IF Q IS SIMPLY CONNECTED AND Q i- C, FOR SOME UNIVALENT f IN H(Q) , f(Q) = 1U. [ 8.1.2 Remark. The result 8.1.1 was stated by Riemann. It was first proved by Koebe who created an algorithm for constructing a sequence {fn } nEN of univalent functions in H (Q) . He showed that for some univalent f in H (Q), fn � f on each compact subset of Q and f(Q) = 1U. Riemann's Mapping Theorem is frequently called the Conformal Mapping Theorem. The term conformal refers to the fact that the mapping f preserves angles (v. 8.1.6, 8.1.7). The PROOF below consists of the crucial 8.1.3 LEMMA followed by the main argument. The line of proof is nonconstructive (ex istential) and is based on the Arzela-Ascoli theme as expressed by Vitali's Theorem (5.3.60). Other tools in the argument are Schwarz's Lemma (7.2.33) and the functions cPa used in the study of Blaschke products (v. 7.2.29-7.2.31 and 7.2.36) .] PROOF. 8.1.3 LEMMA. IF Q IS SIMPLY CONNECTED AND Q i- C, FOR SOME UNI VALENT g IN H (Q) , g(Q) c 1U. 336
337
Section 8.1. Riemann's Mapping Theorem
PROOF of 8.1.3. If (C \ flt i- 0, even if fl is not simply connected, for some b and some positive r, D(b, rt c C \ fl and 9 : fl 3 z f-t � meets z-b the requirements. On the other hand,
fl �f C \ (
-
00 , 0] =
{ z : z = Re iO , R > 0,
-Jr
< (J
r and cle r 9 =f
h+a
meets the requirements. 0 PROOF of 8.1.1. Vitali's Theorem (5.3.60) implies that the nonempty family F of univalent maps of fl into V is precompact in the I ll ao-induced topology of Hb(fl), the set of functions bounded and holomorphic on fl. For b fixed in fl, if k E F, then h �f [k - k(b)] E F and h(b) = O. If
�
h'(b) = I h'(b) 1 e i O and 9 �f e- io h, then 9 E F and g'(b) > O. Hence atten
tion is focused on the nonempty set 9 of functions 9 such that: 9 is univalent in fl , g( fl ) c V, g(b) = 0, and g ' (b) > O. If M = sup { g' (b) : 9 E 9 }, for some sequence {gn } nEN contained in 9, g� (b) t M. Hence, for some subsequence, again denoted {gn } nEN' and some 9 in Hb(fl), g n � 9 on each compact subset of fl and g� (b) t M. Furthermore (v. 5.3.35) , nlim g�(b) = g'(b) = M < 00 The next arguments show that: a) 9 is univalent, whence 9 E 9; b ) g(fl) = V. a) . Since each gn is univalent, M > O. If Zo E fl, for each n, ..... =
.
Chapter 8. Conformal Mapping
338
Hurwitz's Theorem (5.4.39) implies
g(z) - g (zo ) == or [g (z ) - g (zo ) ] l n\{zo} i- 0.
°
If g(z) == g (zo ), then g ' (b) = 0 < M, a contradiction. Hence, for any Zo in Q, [g(z) - g (zo ) ] l n\{zo} i- 0: g is univalent. b) If lei < 1 and e tJ. g(Q), the simple connectedness of Q enters the
g(z) - e is 1 - cg (z ) well-defined and in H (Q) . Direct calculation shows G is univalent on Q and G ( Q ) C 1U. G(z) - G(b) , then H E 9 . Since Finally, if H : Q 3 z f-t cPO ( b) (Z) = 1 - G ( b )G (z ) Ii-:::I 2 Ii-:::I , lei (1 - y l e i ) = 1 - 2 y lei + lei > 0, H (b) = G'(b) 2 = 1 +fi:I ' M > M, 1 - I G (b) 1 2 y lei
argument again: 5.5.19 implies G : Q 3 z f-t JcPc ( g (z ) =
a contradiction.
0
8.1.4 DEFINITION. WHEN (J E [0 , 27r) , THE straight line through a at incli nation (J I S L( a, (J) cle=f { z : z = a + te t° O , t E }
ffi. .
8.1.5 Exercise. If L is a straight line in C and a E L, for a unique (J in [0 , 27r) , L = L ( a, (J). 8.1.6 Exercise. (Conformality, first version) If
f E H (Q) , a E Q, J'(a) i- 0, and L �f L (a, (J) : a) For t in [0, 1] , the equation /' (t) = f (a + te i O ) defines a curVe through f(a) . b) For the line L (f(a), cP) , tangent to /' * at f(a),
cP - (J E Arg [J'(a)] . c) For two differentiable curve-images intersecting at a (whence their f images intersect at f (a)) , the size of the angle between their tangents at a and the size of the angle between the tangents to their f- images at f (a) are the same. [Hint: b) The chain rule for derivatives applies to the calculation of /,' . ]
339
Section S.l. Riemann's Mapping Theorem
8.1.7 Exercise. (Conformality, second version) If
f �f u + iv E H (Q) , J'(a) i- 0, and "/ �f x + iy is a differentiable curve such that ,,/(0) = a: a) x'(O)e l + y'(0)e 2 is a vector parallel to the tangent line at ,,/(0); cle b) for U cle =f u 0 "/ and V =f v 0 ,,/,
U'(O) = u x (a)x' (O) + u y (a)y' (O) , V'(O) = v x (a)x'(O) + vy (a)y'(O);
)
c) U'(O)e l + V'(0)e 2 is a vector parallel to the tangent line at f 0 ,,/(0); u (a (a d) for some ¢ in [0, 27r) , the matrix V x a ) u y a ) is a multiple of the (orthogonal) matrix
( -coSlll� ¢,A-/-,
( x( ) Slll � ) ; cos ,/-,
Vy ( )
e) the vector U'(O)e l + V'(0)e 2 is the vector x'(O)e l + y'(0)e2 rotated through an angle of size ¢.
[Hint: d) The Cauchy-Riemann equations apply.] The two versions of conformality may be reworded as follows. If f E H (Q) and f is invertible at a, f preserves angles at a and the sense of rotation at a. Central to the phenomenon are the Cauchy-Riemann equations, i.e., the differentia bility of f. The material in 10.2.46 is related to the current discussion. 8.1.8 THEOREM. FOR A REGION Q, THE FOLLOWING STATEMENTS ARE LOGICALLY EQUIVALENT: a) Q AND 1U ARE HOMEOMORPHIC; b) Q IS SIMPLY CONNECTED; c) IF a E Coo \ Q AND "/ IS A LOOP FOR WHICH "/* C Q, Ind -y(a) = 0; d) Coo \ Q IS CONNECTED ; e) IF f E H (Q) , FOR SOME SEQUENCE {Pn } nEN OF POLYNOMIAL FUNC TIONS, Pn � f ON EACH COMPACT SUBSET OF Q; f) IF f E H (Q) , "/ IS A RECTIFIABLE LOOP, AND "/* C Q, THEN
i f dz
= 0,
i.e. , IF "/ IS A RECTIFIABLE CURVE AND "/* C Q, THEN
i f dz DE
PENDS ONLY ON ,,/(0) AND ,,/(1) AND NOT ON THE PARTICULAR CURVE "/;
Chapter 8. Conformal Mapping
340
g) IF 1 E H(Q) AND 0 tic I(Q) , FOR SOME F I N H (Q) , F' = I; h) IF 1 E H(Q) A N D 0 tic I(Q) , FOR SOME G I N H (Q) , 1 = exp(G) ; i) IF 1 E H (Q) , 0 tic I(Q) , AN D n E N, FOR SOME H I N H (Q) ,
I = Hn .
[ 8.1.9 Remark. Listed below are implications already estab lished and their provenances. These implications and their deriva tions are the root of the subsequent argument. The entire set is intimately related to 8.1.1.] Implication b) '* d) b) '* e) b) '* f) b) '* g) b) '* h) b) '* i) PROOF. a)
'*
Provenance 7.1.21 7.1.19 5.5.16c) 5.5.17 5.5.18 5.5.19
b) : If "( is a loop, "(* C 1U, and J(t)
F(t, ) clef = S"((t)
==
0, then
s
is a homotopy such that "( '" F,1U J. If \11 : 1U r-+ Q is a homeomorphism and f r is a loop such that r* c Q, then \11 - 1 0 r � "( is a loop such that "(* C 1U and \11 0 F 0 \11 - 1 �f is a homotopy. Furthermore, if Ll(t) == \11 ( 0) , then r
Ll.
b) '* c) : v. 5.5.16c ) . c) '* d): v. the PROOF of 7.1.21. d) '* e): v. 7.1.14. e) '* f): Every polynomial is a derivative and f) obtains for derivatives. f) '* g): If { a, z} C Q, there is a polygon n connecting a to z and a corresponding "( such that "(* = n . For the map "' ''' , n
f
F : Q '3 z r-+ F(z) �
11 I b (t)] d"((t) ,
F'(z) = I(z). Owing to f), F(z) is well-defined, Le., is independent of the choice of "( so long as "((0) = a. f' f' g) '* h): For some F, F' = 7 ' If clef = exp(F) , then ' = 7 whence '
(
0 is
: h E F} .
In effect, 8.1.13 restates the Riemann Mapping Theorem as a result in the calculus of variations. Lemma 8.1.3 provides a motivation for Rie mann's original attempt to prove 8.1.1. According to 4.7.18, if f : Q r-+ 1U is univalent and holomorphic, the Cauchy-Riemann equations imply that the area of f(Q) is
A[J(Q)] =
�f in 1 f' (z) 1 2 dx dy
in (u; + u�) dx dy = in (v; + v�) dx dy ::::: 1.
Hence, if A[J(Q)] is maximal, e.g., if A[J(Q)] = I, f is a good candidate for the biholomorphic map of Q on 1U. Euler's equations for the stated vari ational problem take the form t:m = v = 0 and the boundary conditions for and v are simply that for all a in 8(Q), lim l (z) 1 lim I v(z) 1 ::::: 1. The z=a z = aregion, for some discussion in 6.2.17-6.2.22 implies that if Q is a Dirichlet 0, lim l u(z) I , i.e., Dirichlet ' s problem has a solution. z=a
u u, l1u ln= u la( n) =
l1
u V
Chapter 8. Conformal Mapping
342
Riemann's proof of 8.1.1 involved the implicit assumption that there is a solution to the problem of maximizing (u ; + u�) dx dy subject to
In
In
the normalizing condition, (u; + u�) dx dy � 1. The assertion that this kind of variational problem has a solution became known as Dirichlet 's
Principle.
However there appeared 8.1.14 Example. (Weierstrafi) Among all 1 in C OO (lR, lR) such that I(x) = 0 if I x l ::=: 1 and 1'(-0.9) = -1 '(0, 9) = 1 there is none for which
11 { t2 + [J' (t)] } 2
�
dt is least. (The problem is to minimize the length of a curve "( : [0, 1] '3 t r-+ t + i l (t) such that "((0) -"((I) = -1, "( t o. The =
geometry of the situation shows that the infimum of all such lengths is 2 but that the length of each such curve exceeds 2.) Thus Dirichlet's Principle, as a statement about the solvability of a variational problem, was suspect. In the hands of (alphabetically) Hilbert, Koebe, Konig, Neumann, Poincare, Schwarz, Weyl, and Zaremba, the validity of Dirichlet's Prin ciple for a simply connected fl achieved a semblance of validity. In mod ified form, Dirichlet's Principle is central to one of the derivations of the Uniformization Theorem, v. 10.3.20. Given the validity of Dirichlet's Principle for a si�ply connected fl, Riemann ' s approach leads to a u and a harmonic conjugate v such that 1 u + iv is the required conformal map of fl onto 1U. The intimate connection of Dirichlet's Principle to the solvability of Dirichlet's prob lem seems to bind the two to questions about Dirichlet regions, barriers, simple connectedness, etc. The PROOF of 8.1.1 resolves these questions. The discussion in Section 8.5 is also germane to the considerations above. For the harmonic function g corresponding to a Green's function G(·, a), v. 8.5.1 -8.5.9, some harmonic conjugate, say h, of g leads to a function ¢ h + i g E H (fl) . If, for z in fl, I(z) (z a) exp[¢(z)] , then is a conformal map of fl onto 1U and I ( a ) o. 1
�f
�f
�f
-
=
8.2. Mobius Transformations
If Z; , 1
:::; i :::;
4 , are four elements of C, the number
is their cross ratio or anharmonic ratio. For Z2 , Z3 , Z4 fixed and pairwise different, X (z, Z2 , Z3 , Z4 ) M(z) is a function on Coo \ { Z3 } and may be
�f
Section 8.2. Mobius Transformations
00 There are constants a,
extended to Coo by defining M (Z3) to be a z + b clef such that M(z) Tabed(Z) and d
= cz =
.
+
343 b, c, d
More generally, when l1 -j. 0, the map
Tabed : C '3 z r-+
--cz az + b d +
is a Mobius transformation. By definition,
( - � ) 00 and Tabed(oo)
Tabed Correspondingly, for
e
=
as
=
�.
in Section 5.6, there is
� clef e - 1 T"b(.d8 : L: 2 \ { (0, 0, 1) } r-+ L: 2 , Tabed =
L: 2 •
which may be extended by continuity to a self-map of When ambiguity is unlikely, the subscript abed is dropped. Note that if a -j. 0, then
=
1 whence, if a V75. ' then (a a) (ad) - (ab) (ac) = 1, and as the need arises the value of l1 may be taken to be 1.
8.2.1 Exercise. Each Mobius transformation T is invertible and
:
-
-1 C '3 z r-+ dz + b = T(- d) be (- a ) . Tabed cz - a (Thus each T is one-one: {T(z) T (z') ) {} {z = z' } . )
=
8.2.2 Exercise. a) The set of all Mobius transformations Tabed is a group with respect to composition 0 as a binary operation. b) Those for which l1 = a d - bc 1 is a normal subgroup M 1 contain ing the normal subsubgroup E �f { TW01 ' T(- l )OO(- l ) }. c ) The quotient group M d E, denoted is isomorphic to SL(2, q , the multiplicative group of all 2 x 2 matrices M with entries from C and for which det (M) 1.
M
=
=
Mo,
344
Chapter 8. Conformal Mapping
zP
I w - Z l 2 = I Z - a l · I ZP - Z l = I Z - a l · I ZP - a l - I Z - a l 2 = I Z - a l · I ZP - a l - I W - ZI 2 ) , I Z - a l · I ZP - a l = r 2 . -
(r2
Figure 8.2.1.
When r > 0, z -j. a, and k clef =
1
2
r z-a
1 2 ' the pomt .
zP clef = a + k(z - a) is the reflection or inversion of z in Ca(r) �f a [D( a, r rl and z is the reflec tion of z P in Ca(r): z = (zP) P . By abuse of notation, aP = 00 and ooP = a. Figure 8.2.1 above illustrates the geometry of reflection or inversion in the circle Ca(r). For a line L( a, 0) the reflection of z � a + z - a l in L( a, 0) is
1 Z P clef = a + 1 z - a 1 e - i ( q, - 2 0 )
eiq,
and z is the reflection of z P in L( a, 0) . 8.2.3 Exercise. The reflection of z in the line L (O, O) IS z . For L(a, O) regarded as a mirror, zP is the mirror image of z.
345
Section 8.2. Mobius Transformations
The superscript P serves as a generic notation for a reflection z r-+ zP performed with respect to some circle or line. 8.2.4 THEOREM. IF z E C AND ad be -j. 0, THEN Tabcd(Z) ARISES FROM THE PERFORMANCE OF AN EVEN NUMBER OF REFLECTIONS (IN LINES OR CIRCLES) . -
PROOF. The argument can be followed by reference to Figure 8.2.2. For any z, TOl lO (z) arises by reflecting z in '][' and reflecting the result in JR. If b -j. 0 there are (infinitely many) pairs of parallel lines L 1 , L 2 , sepa and perpendicular to the line through b and o. Direct calcu rated by lation reveals that T1 b0 1 (Z) arises by reflecting z in L 1 and reflecting the result in L2 •
I�I
Z1: the reflection of z in L1
z12: the reflection of z1 in L2
o
Figure 8.2.2.
346
Chapter 8. Conformal Mapping
If ° :::; (J < 27r, then TeiO OOl (z) arises by reflecting z in L3 and reflecting the result in L4• If ° < A E lR, then TA00 1 ( Z ) arises by reflecting z in '][' [= Co (I)] and reflecting the result in Co Finally,
(VA).
T.abed (z) -
{
r::.
be - ad + ( z d) if e -j. ° e ee + b a if e = O z + d d
D
8.2.5 Exercise. The elements (z, z') in C� are a pair of mutual re flections in a circle C resp. a line L iff X (z', Z I , Z2 , Z3 ) = X (z, Z I , z2 , Z3 ) is meaningful, , i.e., # (z', Z I , Z2 , Z3 ) # (z, Z I , Z2 , Z3 ) = 4, and true. =
8.2.6 Exercise. The validity of X (z', Z I , Z2 , Z3 ) = X (z, Z I , Z2 , Z3 ) is inde pendent of the choice of an acceptable triple Z I , Z2 , Z3 . 8.2.7 Exercise. If T E or 2.
M \ {id }, the number of fixed points of T is 0, 1 ,
1 . sgn a 8.2.9 Exercise. If {Z I ' Z2 , Z3 } resp. { W I , W 2 , W3 } are two sets of three points in Coo , for precisely one T in T (Zi ) = Wi , 1 :::; i :::; 3. 8.2.10 Exercise. (Extended Schwarz Reflection Principle) If: a) r > 0; b) A dcl = { a + ret 0 :::; (J l < (J < (J2 :::; 27r } ; c) 8.2.8 Exercise. If ° < 10'1 < 1 the fixed points of 4;" are ±
--
Mo,
e
:
1 E H [D(a, rt] n C [D(O, atl:.JA, q ;
d) 1 (A) C Cb ( R) ; e) z r-+ zPa resp. z r-+ ZP b is the map z r-+ zP performed with respect to Ca(r) resp. Cb ( R) ; f)
F(z) then Q
�f
{
if I z - a l < r I( z) [J (zPa Wb if I zPa - a l < r , if z E A; I(z)
�f D(a , rtl:.JAl:.J [D(a, rtJ Pa is a region and F E H (Q) .
8.2.11 Exercise. If K is a circle lying on L 2 , then 8 (K \ { (O, O, I)}) is a circle or a straight line. 8.2.12 Exercise. The group is generated by the subset
M
1
To : C \ {a} '3 z r-+ , Tab : C '3 z r-+ az + b, a, b E C. z -
Section 8.2. Mobius Transformations
347
8.2.13 Exercise. a) If .c is the set of all circles and straight lines and then T(.c) = .c. b) If D is the set of all open discs and the comple ments of all closed discs and T E then T ( D) = D. 8.2.14 Exercise. If a, b , c, d E lR and ad - bc > 0, then Tabed leaves n + invariant: Tabed (n + ) = n + .
TE
M,
M,
8.2.15 DEFINITION. FOR A REGION Q, Aut (Q) IS THE SET OF CONFOR MAL AUTOJECTIONS OF Q. 8.2.16 Exercise. If Q is a region, with respect to composition 0 as a binary operation, Aut (Q) is a group. 8.2.17 Example. According to 8.1.10, Aut (V)
{ T : T(z) = e iO ¢,,(z ) ,
=
a
E V, 0
M.
-::;
(J
L l¢n (z) 1 2 :::; 1J2 n=1 7r
n
58 .
E 58 AND {¢n } nEN IS A CON IN H (Q), THEN
E N, no further argument is needed.NThat possibil ity aside, for some N, the map eN '3 � (a l , . . . , a N ) r-+ L an¢n (z) is a PROOF. If ¢n(z) = 0,
a
n= 1
Chapter 8. Conformal Mapping
352
continuous open map of the Banach space eN onto C. Hence, for some a, N L an q)n (z) = 1, and A �f { f : f E span ( q)1 , . . . , q)N ) , f (z) = 1 } -j. (/). If
n= l N N N f E A, then f L (I, q)n ) q)n �f L cnq)n and so 1 = L cnq)n( z). n= l n= l n= l Schwarz's inequality in the current context takes the form =
1�
N
N
L I cnl 2 L lq)n (z) 1 2 n= l n= l •
•
( 8.3.13 )
Since ( 8.3.13 ) is valid for all large N, direct calculation yields the conclusion.
D
8.3.14 COROLLARY. THE SERIES ( 8.3. 1 1 ) CONVERGES ABSOLUTELY IN
Q.
a
PROOF. If E Q there are sequences
bers such that: •
•
•
•
g
=
{an } nEN ' { ;3n } nEN of complex num
lanl = 1 ;3,,1 1, n E N; {anq)n } nEN is a CON in SJ (Q) ; anq),, (a) = l q)n (a) l , n E N; (;3nf, anq)n ) 1 ( 1, q)n ) l· =
=
The Fischer-Riesz Theorem (3.7.14) implies that for some g in SJ ( Q ) , 00 L (;3n f, anq)n) anq)n and dn �f (g , anq)n) 1 (I, q)n ) l , n E N.
n= l
=
S,h=,,', inequality implie, 8.3.15 Exercise. The series
00
n= l
(� I (t, ¢,,) I ' 1 ¢,, (a) I ) '
No, p E C and G (pt -j. 0. If q E G(pt , then p E G ( q t . ] 8.4.4 Exercise. a) When G is a properly discontinuous subgroup of Mo and p and N(p) are the objects in 8.4.1, for some nonempty subset 5 of Mo, if S E 5, then S(p) = 00. b) If S E 5 the set r �f SGS - 1 �f { STS- 1 : T E G } �f { T
}
is a properly discontinuous subgroup of Mo and is isomorphic to G. c) If Tabed E (r \ {id }), then Tabcd( (0 ) -j. 00 and c -j. O. [Hint: c). If 00 T (oo) , then p E [Coo \ N(p)] .] =
Below, each properly discontinuous group, however denoted-r or G-is assumed to conform to c) in 8.4.4. 8.4.5 LEMMA. IF p AND N(p) ARE THE OBJECTS IN 8.4.1, FOR SOME POSITIVE p, IF I z l > p AND T E (r \ {id } ) , THEN T( (0 ) -j. z. PROOF. If Tn E r \ {id } , Tn (oo) �f Zn , and I Zn l > n, for some Rn in G, Tn = SRn S- 1 , Zn = SRn S- 1 (00) SRn (p) , S - 1 (zn ) = Rn (P) E [Coo \ N(p)] , 1 S- (zn ) = S - I (oo ) = P E [Coo \ N(pW = Coo \ N(p) , nlim ---+ oo =
Section S.4. Groups and Holomorphy
359
a contradiction.
D
8.4.6 Exercise. If Tabed E Mo, then T�bed (z) =
1
(cz + dF
.
8.4.7 DEFINITION. FOR A GIVEN Tabed IN Mo, WHEN c -j. 0 THE SET { z : I cz + d l = I } IS THE isometric circle Cabed OF Tabed:
( -�, � )
THE ( CLOSED ) DISC D ed. WHEN r IS I I IS THE associate OF Tab A PROPERLY DISCONTINUOUS SUBGROUP OF Mo,
clef AND C =
{ - d : Tabed E r } . �
8.4.8 Exercise. a) The isometric circle of T;;;'�d is the image under Tabed
of Cabed. b) If
Tabe d E (r \ {id }) and z E Coo , then Tabe d(Z) arises by inversion of z in Cabe d, a reflection in L, the perpendicular bisector of the line joining - -dc and -ac , and a (possibly trivial) rotation centered at � . c [Hint: For a), if z E Cabe d, for some 8, z=
-d + e i O and Tabed(Z) c
a _ e- i O c
= ---
For b) , 8.2.4 applies.] 8.4.9 Exercise. a) 00 E R r ; b) If T E r \ {id } the radii of the isometric circles of T and T- 1 are equal. 8.4.10 Exercise. If {S, T} c r, ST- 1 -j. id , {CS , CT, CST, CS �l , CT �l } is the set of centers, and {rs, rT, rsT, rs�l , rT�l } is the set of radii of the isometric circles of { S, T, ST, S- 1 , T - 1 } : a) rT2 rSTrT = -- = rST C rSl rT c l e ST TI rs I CT� l - cs l .' I T� - cs l ' b)
=
l esl < p; c) rs < 2p.
360
Chapter 8. Conformal Mapping
The union of all the discs that are associates of elements of r is contained in a disc of finite radius, say D(O, A) . If I z l > A, then z E Rr. [Hint: b): If I cs l
>
p, then 5- 1 (00)
8.4.11 THEOREM. a) THE UNION U
=
cs ; v. 8.4.5.]
�f Ur T (Rr) �f r ( Rr) IS DENSE IN TE
C. b) IF u E Rr AND T E r \ {id } , THEN T(u) tic Rr. c) I F u E 8 ( Rr), N(u) I S A NEIGHBORHOOD OF u, AND v E N(u), FOR SOME W I N Rr AND SOME T IN r \ {id } , T(w) = v . PROOF. a) If U is not dense in C, for some positive r and some u, D(u, rt
C
Coo \ u.
Furthermore, for each T in r, T [D(u, rn T [D(u, rt]
c
C
Coo \ U. In particular,
C \ Rr.
Since 00 E Rr, the center of each isometric circle is in r ( R r), whence u is not the center of any isometric circle. On the other hand, since u tic R r, u is in some D , the associate of some Tabe d. Furthermore, 8.4.8 I I implies that Tabed arises by a reflection in Cabed followed by a reflection in
(- �, � )
a line and a possibly trivial rotation (a pair of reflections in lines) . If z E Cabed and the radius of Tabe d [D(z, r)] is R, direct calculation r shows R = 1 - c 2 r2 '
11
(
)°
Since r < � < 2p, if z E D - �d , �1 , then 1
R
> ----,2,.-
r
r 1-4 2 p
cle=f kr
>
r.
-
Hence the radius Rm of T;bed [D ( z, r)] exceeds k Tn r, m E N. If m is large, T;be d [D(z, r)] meets Rr, a contradiction. b) If T E r \ {id } , since u is not in the associate of T, 8.4.8 implies T(u) is in the associate of T- 1 , hence is not in Rr. i.e., c) By definition, N (u) meets some Coo \ D _
(
)
( � , I �I )
d 1 ° N(u) n D - � ' � = 0.
0,
361
Section S.4. Groups and Holomorphy
If D ( �e ' �) l e l is the associate of Tabed, 8.4.8 implies -
D
8.4.12 LEMMA. IF S AND T ARE TWO ELEMENTS OF r, THEN T (R r ) n S ( R r ) = 0.
PROOF. Since S - I T E r
\ { id } , 8.4.11b) implies S - I T (R r) n Rr = 0.
D
8.4.13 DEFINITION. FOR A PROPERLY DISCONTINUOUS GROUP r, A FUNC TION f IN M (UO ) IS r-automorphie IFF FOR EACH T IN r, f T = f.
) )
0
8.4.14 THEOREM. ( Poincare a IF Rl IS A RATIONAL FUNCTION, P (Rd n (et = 0, AND N '3 m > 2,
THEN ON EVERY COMPACT SET DISJOINT FROM P( R) u e· ,
CONVERGES UNIFORMLY AND DEFINES A FUNCTION (h IN
{ Coo \ [P (R I ) u e·l } . b ) IF Tabed E r, THEN (h [Tabed(Z)] = ( ez + d) 2rn (h ( z ) . c) IF (h CORRE SPONDS TO A RATIONAL R2 , RESTRICTED LIKE R1 , AND, FOR Z NOT IN (h ( z ) , THEN F IS r-AUTOMORPHIC. Z (t'h ) u P (Rd u e· , F ( z ) cle=f (h ( z ) PROOF. a ) Since the set e of the centers of the discs associated to the H
--
elements of r is bounded, and since the set of radii of those discs is also bounded, there is a positive number p such that for each Sa(3'Yii in R r ,
The circle a
[D ( � ) ] �f Cabed is concentric with Ca(3'Yii ' -
,p
362
Chapter 8. Conformal Mapping
All the associated discs are contained in each D the complement of each D
( -� , p) , whence
( -�, p) is contained in R r .
If Tabed E Rr, then Tabed (Cabed ) results from an inversion in Cabed fol lowed by reflections in lines. The reflections in lines are isometric maps, i.e. , they do not alter distances between points. The inversions in circles multi ply distances between points by a constant dependent on I c l (v. PROOF of 8.4.11). Thus the radius of Tabed (Cabed) is p 2 ' l 1 The complement of Cabed consists of a bounded open disc and an un bounded component. The unbounded component is mapped by Tubed onto the bounded open disc determined by Tabed (Cubed) . The bounded open disc is a subset of Tabed (Rr). Hence, owing to 8.4.12, if SOi(3'Yii and Tabed are two maps in Mo, the interior of the intersection of the bounded open disc determined by S"(3'Yii (COi(3'Yii ) and that determined by Tabed (Cabed) is empty. It follows that
�
(8.4. 15) .
If K is a compact set disjoint from P(R) u e· , for some constant M,
The statements b) and c) follow by direct calculation.
D
R1 is not a constant, then F is not a constant. 8.4.16 Exercise. If _ R2 When G is a properly discontinuous group, unrestricted by the conditions in 8.4.4c), a fundamental set G in C is a set for which � i- (/) and the statements 8.4.11b) -c) (with r and R r replaced by G and G) obtain. In particular, Rr is a fundamental set for r. [ 8.4.17 Note. In GW1 , W2 or Mod, there are elements T different from id and for which T' = 1 . For such T there is no corresponding
isometric circle and the associate of T is meaningless.]
,
8.4.18 Exercise. If G is an arbitrary properly discontinuous group and, in the notations introduced above, r = S GS - 1 then S - 1 R r serves as a fundamental set G for G. 8.4.19 Exercise. a) If G = GW1 , W2 ' the interior of the parallelogram P determined by O, W 1 , W2 together with [0, w I ) U [0, W2 ) is a fundamental
363
Section 8.5. Conformal Mapping and Green's Functions
{
-�
�,
},
set G . b) If G = Mod, A �f z 'S (z) > 0 B is � �(z) < the union of the complements of the interiors of all meaningful associates 27r of elements of G, and C cle=f z : Z = ei O ' 2"7r � (J � 3 , the union of
{
:
}
D �f (A n B) U C and its reflection DP in lR is a fundamental set G . • •
When G is unrestricted, there are the following options . One can work with r �f SGS - 1 for some appropriate S and apply the machinery developed above. One can stick with G and cope with complications that can arise when the set of centers of meaningful isometric circles has a cluster point at 00 .
Sometimes, e.g., when G = GW1,W2 ' there are no isometric circles, and the resulting discussion is straightforward. The emerging theory is that of elliptic functions to which SOme of the most important contributions came from Weierstrafi [ Hil] , v. 8.6.1 1. For some properly discontinuous groups r , detailed elaboration of the reasoning behind 8.4.14 leads to the construction [Ford] of r-automorphic functions enjoying special properties, e.g., having in R r exactly one pole of order one and one zero of order one. Manipulation of such functions leads to others that are holomorphic in n + and map n + onto C \ {O, I}. Such functions can be used to prove the Little Picard Theorem (9.3.1) [Rud] . 8 . 5 . Co nformal Mapping and Green's Functions
As noted 6.5.1 if h E H(Q) and 0 tic h(Q) , then In I h l E HaJR(Q). When Q is simply connected and f Q r-+ 1U is a conformal map, for some a in Q, f(a) = 0, whence G( · , a) �f - In If I is not defined at a but is defined in :
Q \ {a}.
Q , a, AND f ABOVE: a) G( · , a) E HaJR (Q \ {a}); b) IF b E 8oo(Q) , zlim G(z, a) = 0; c) FOR SOME POSITIVE r , ---+ b g Q '3 z r-+ G( z, a) + In I z - a l IS IN HaJR [D(a, rt l. 8.5.1 LEMMA. IN THE CONTEXT OF
:
PROOF. a) The Hint in 6.5.1 applies.
364
Chapter 8. Conformal Mapping
b) If fl '3 Z -+ b E 8oo (fl) and In I f(z) 1 -1+ 0, via passage to subsequences as needed, for some 15 in (0, 1), and some {zn} nEN ' fl '3 Zn -+ b while f (zn ) converges to some d in 1U and If (zn )1 � 1 - 15. For the sequence
of compact sets, Km
C
K� + 1 and
U Km = mEN
1U.
phic nature of f, each f - 1 ( Km ) is compact and
Owing to the biholomor
Furthermore, 1.3.7 implies that for some {wn} nEN , W n -+ C � f- l (d) and f (wn ) == f (zn). For some large m and some large n, Wn E f - 1 (Km ) and Zn E fl \ f - 1 (Km ) , whence f is not bijective. - a is in H (fl). c) The function fl '3 Z r-+ zf(z) D
a
a)
8.5.2 DEFINITION. FOR A REGION fl AND AN IN fl, A FUNCTION G(·, CONFORMING TO a) -c) IN 8.5.1 IS (A) GREEN'S FUNCTION FOR fl. Riemann's Mapping Theorem (8.1.1) and 8.5.1 imply that for a sim ply connected proper subregion fl of C and a point in fl, (a) Green's
a
function G(·, a) exists. The following items delimit to some degree the kinds of regions for which there are and are not Green's functions. 8.5.3 Exercise. If fl is a bounded Dirichlet region and a E fl, there is a Green's function G(·, a) for fl. [Hint: The solution of the Dirichlet problem for the boundary condition u la ( n / z) = In I z - a l serves.] 8.5.4 Exercise. If fl
for fl.
=
1U \
{o} � 10 there is no Green's function G(·, O)
[Hint: The discussion in 6.3.11 applies.]
a)
8.5.5 Exercise. If G(·, is (a) Green's function for fl, then G > 0. [Hint: The Maximum Principle applies.] 8.5.6 LEMMA. THERE IS NO GREEN'S FUNCTION FOR C. PROOF. If G is a Green's function for C, E > 0, r > I Z2 -
{I z - zl l < r5} ::::} {IG(z) - G(zI )1 < E} ,
zl l > 15 > 0, and
Section 8.6. Miscellaneous Exercises
365
G (Z I ) + E (ln z - z - ln r) I. S harmomc. m . C \ {zd · Furthen gr ( z ) cl=e f (ln ll J - ln r) l thermore, if A �f A (z l ; J, rt, on 8(A) �f r �f CZ 1 (J) u CZ 1 (r), G l r :S: gr l r . The Maximum Principle implies G I A :S: gr I A · As r t 00 there emerges i.e., G (Z I ) :s: G (Z2 ) : G is a constant and cannot be a Green's functi�n.
D
8.5.7 Exercise. a) If G ( · , a) is a Green's function for Q , then G ( · , a) is unique, and if O(z, a) + In I z - a l is harmonic near a, then 0(. , a) > G. [Hint: The Maximum Principle applies.] 8.5.8 Exercise. If f : Q 1 r-+ Q2 is a biholomorphic bijection and G2 (·, a) is the Green's function for Q 2 , then G2 (·, a) o f is the Green's function G1 [ . , J - l (a)] for Q l . [Hint: The Open Mapping Theorem applies.] 8 . 6 . Miscellaneous Exercises
8.6.1 Exercise. If T E Mo and p, q , r, s are four complex numbers, then
X[T(p), T( q), T(r), T(s)] 8.6.2 Bxercise. a) If 5 E Mo, and 5 - 1
=
X(p, q, r, s).
{O, 00, 1} = {Z2 , Z3 , Z4 }, then
b) Four points p, q, r, s are cocircular or collinear iff X (p, q, r, s) is real. P-q . · · p, q , r are co llmear 8 . 6 . 3 ExerClse. Three pomts 1· ff -- IS. rea1 .
q-r Q, and 5· n Q = 0: a) Q \ 5 is a
8.6.4 Exercise. If Q is a region, 5 c region; b) SJ(Q) = SJ (Q \ 5). [Hint: For b) the argument for 8.3.2 applies.]
If Q is a region, S ,B '3 5 c Q, Q \ 5 IS a region, and )' 2 (5) = 0, then SJ(Q) = SJ (Q \ 5). [ 8.6.6 Note. If 5 = CO! (some Cantor set contained in [0, 1]), ). 2 (5) = ° and 1U \ 5 is a region.
8.6.5 Exercise.
If 5 �f (- 1 , 1 ) , ). 2 (5) = ° but 1U \ 5 is not a region.
366
Chapter 8. Conformal Mapping
=
If 5 �f { z : z p + iq, {p, q} tains no region.]
C
QI } , >' 2 (5)
=
0,
but 1U \ 5 con
8.6.7 Exercise. If Q { x + iy : I x l < 1, I y l < 1 } what is the corre sponding Bergman kernel K? [Hint: The Gram-Schmidt algorithm applies to the sequence
�
8.6.8 Exercise. If Q 1
Q2 cle=f
�f { z
{z : z
:
=
0 :::;
a < '25(z) < b :::; 27r } and
Re iO , 0 < R, a < (J < (3 } ,
for some real c, h Q 1 '3 z r-+ eic z is a conformal map of Q 1 onto Q2 . 8.6.9 Exercise. If the Schwarzian derivative, :
{w , z }
(=
)
2w'(z)w "'(z) - 3 [W " (Z)] 2 ' 2 [w ' (z)] 2
is regarded as a function of w and z, and T E Mo, then {w, z} = {T(w), z }
=
[T(z) '] 2 {w, T(z) } .
8.6.10 Exercise. When {5, T} C Mo: a) If 5 has only one fixed point while T has two, then 5T -j. T5; b) If 5 and T have the same fixed point ( s) , then 5T T 5. c) If 5T T 5 and each of 5 and T has only one fixed point, they share it. 8.6.11 Exercise. a) If =
=
� P ( Z l �l , �2 ) �f z12 L +
{
�}
I then P , the _ 2- 2 ' 3w,tO [Z W] -automorphic. b) If f is G Weierstrafl elliptic function, is G automorphic and 5 � { a + + : 0 :::; s, < I } is the period paral lelogram vertexed at a: bl) the sum of the residues of the poles of f in 5 is 0; b2) 2 :::; #[P( f ) n 5] = #[Z (f) n 5] < 00.
and
P
Q
W 1 , W2
tWl SW2
t
W 1 , W2 -
Section 8.6. Miscellaneous Exercises
8.6.12 Exercise. If _ef 60 g2 �
p' p
" and g3 cl=ef 140 " -:-----� � [Z ] 3w,tO 3w,tO [Z - w] 1
Q
367
then = 4 3 - g2 P - g3. 8.6.13 Exercise. If �f W
- W
4
1
Q
6 '
m l Wl + m2w2 -j. 0, then
(
(�l )
;
)
�)
8.6.1 4 Exercise. If �f P , e2 �f P WI W2 , e3 �f P ( 2 , then the ei , 1 :::; i :::; 3, are the three zeros of the polynomial function Z r-+ 4z 3 - g 2 Z - g3 .
el
a la p- ( )]
8.6.15 Exercise. If E Coo and multiplicities are taken into account, for = 2. P as in 8.4.19, # [p n The statements in 8.6. 16 below are steps leading to the sufficient con dition for the existence of a barrier at a point in the boundary 8(Q) of a region Q, 6.3.29. The argument has been deferred to this part of the text because 8.1.8 is used. 8.6. 16 Exercise. If E r �f 800 (Q) and no component of Coo \ Q consists of alone: a) The assumption = 00 is permissible. (Otherwise, for the map v.
a
a
a a
--a 1
¢ : Coo '3 Z r-+ z - , the argument may be conducted on ¢ (Coo ). ) b) If C is a component of Coo \ Q and 00 E C, then Q 1 �f Coo \ C is simply connected, d. 8.1.8d ) , and Q C Q . c) For some f in H (Q I ), exp[ J (z)] = z, d. 8.1 .8h ) (f is a determination of In) . d) If Q2 �f f ( Q ) and L �f + it : E lR, - 00 < t < oo } , the line L meets Q2 in at most count ably many open line segments ( w � , w �) of total length not exceeding 271" so that 'S ( w �) > 'S ( w U . (Otherwise, for some finite K, in Q 2 there are K points Z l , . . . , ZK such that
{a
a
l
K-l L [In (Zk+I ) - In (zk)] = 271"i, k=l
exp (In ZK - In z I )
=
exp(271"i) = 1,
Chapter 8. Conformal Mapping
368
a contradiction.) e) If w �f u + iv, U 2: a there is a holomorphic function ( h such that w� - w = exp [il'h(w)] w� - w
and 0 � fh (w ) f) If
�
7r.
e(w) cle=f
{ -� 7' 7r
'" 1'h (W)
if �(w) 2: a if �(w) < a
-1
and Q3 �f { w : � (w) > a }, then Furthermore,
- -7r2 arctan
7r
� (w ) - a
�
e(w) � o.
(The function f is a (holomorphic) determination of In. Hence the imaginary part of f is harmonic and so is (J. The calculation in e) and plane geometry provide the estimate . ) g) If an t 00 and en corresponds to an as e corresponds to a, then (3(z)
00
� nL 2 - n en (ln z)
=O defines a function subharmonic in Q and tending to zero as z -+ 00 . h) Near any point in r, for all sufficiently large n, the functions en take on the value - l . i) In the sense of Ahlfors and Sario [AhS] , the function (3 is a barrier at
a.
[ 8.6.17 Note. The reason for g) is that e(ln z) can converge to as z converges to some point on r .]
o
8.6.18 Exercise. If Q is simply connected, then Q is a Dirichlet region. [Hint: If Q ¥ C, then Coo \ Q consists of precisely one component: 8.6.16 applies. If Q = C, 8(Q) = 0.]
9
Defective Functions
9.1. Intro duction
E
The set of entire functions is divided into two subsets, namely the set P of polynomial functions and T \ P the set of transcendental functions. If p E P, then p(C) = C is an abbreviated statement of the Funda mental Theorem of A lgebra (FTA). On the other hand, z -+ exp(z) is a transcendental function and exp(C) = C \ {O} . If f T, the Weierstrafi-Casorati Theorem (5.4.3c)) implies that near 00, the values assumed by f are dense in C. Thus zero is an isolated essential singularity of g C \ {O} '3 z r-+ f By abuse of language, 00 is an isolated essential singularity of f. The following result embraces all the phenomena just described.
E
�f E
(�).
:
E
9.1.1 THEOREM. (The Great Picard Theorem) IF a IS AN ISOLATED ES SENTIAL SINGULARITY OF A FUNCTION f, R > 0, AND f H [A (a; 0, Rt] '
THEN # {C \ J [A(a; 0, Rt]} � 1 .
I n every sufficiently small punctured neighborhood o f an iso lated essential singularity of a function f, the range of f omits at most one point.
The discussion is facilitated by the introduction of some special vocab ulary and notation.
E
�f
9.1.2 DEFINITION. WHEN f CIC , C \ f(C) D(f) IS THE SET OF de fections OF f AND f I S # [D(f)]-defective. If and f is 2-defective, say {a, b} C [C \ f(C)] , then
fEE
] �f � = � E E and {O, I} C C \ [ ](C) ] . Similarly, if f E M and f is 3-defective, say {a, b, c}
C
[Coo \ f (Coo )] ,
369
Chapter 9. Defective Functions
370
then I- cl�f cC -- ab . I - ab E M and {a, 1, 00} c [Coo \ I (Coo )] . The study of 12-defective entire resp. 3-defective meromorphic functions may be confined to functions with the simple sets of {0, 1} resp. {0, 1, 00} of defections. Hence, absent any further comment, for an entire 2-defective function I, D(f) = { o, I}; for a meromorphic 3-defective function I, D( f) = {a, 1, 00}. The discussion below is devoted to showing first that 2-defective entire functions and 3-defective meromorphic functions are constants. Elabora tions of those results provide the contents of the Great Picard Theorem. 1 9.1.3 Exercise. If I E M (C) and D(f) = {a , b, c}, then h cle=f _ E E I a 1 _ , _1_ } . (Thus the study of 3-defective meromorphic and D ( h ) = { _ b- a b-c functions is reduced to the study of 2-defective entire functions.) The arguments that follow are an amalgam of the efforts of several writers: Ahlfors, Bloch, Bonk, Caratheodory, Estermann, Landau, Minda, Montel, and Schottky. Picard ' s original proof of his Little Theorem is of an entirely different character, v. [Hil, Rud] . Four steps are involved. The first seems irrelevant to the goal. A. (Bloch) If I E H ( U) and 11' (0)1 2: 1, for some I-free positive constant B, and some b (dependent on I) , B 2: 11 and I (1U) :J D(b, Bt. 2 B. If h is a nonconstant entire function and r > 0, for some b, --
--
h (C)
:J
D(b, rt ·
The range of a nonconstant entire function contains open discs of arbitrary radius. C. If l E E, I is not a constant, and D(f) = {a, I}, associated to I is a nonconstant entire function F such that D(F)
:J
S cle=f
{
± In( Vm +
� V
n7ri
m - 1) + 2
:
m E N,
n E ;Z } .
Since, for any b, D(b, It n S -j. 0, F (C) fails to contain any open disc of radius 1, a contradiction of B: the Little Picard Theorem is valid. D. Refined extensions, due to Schottky, of A together with a special ap plication of the ideas behind the Arzela-Ascoli Theorem, v. 1.6.9, lead to the Great Picard Theorem.
Section 9.2. Bloch's Theorem
371
9.2. Bloch's Theorem
The next rather general result provides an entry to the entire complex of Bloch/Landau/Schottky theorems. 9.2.1 Exercise. If X is a topological space, Y is a metric space, f : X r-+ Y is open, a E V E O(X), and r5 �f inf { d[y, f(a)] : y E 8[J(V)] }, then r5 > 0 and B[J(a), W C f(V). [Hint: 1 .3.7 applies.]
IF r 0, f E H[D(a, r)], M �f 11 !' I D(a,r)o 11 00 :::; 2 1!, (a)l, AND R = (3 - 2v2)r 1!, (a) l , THEN D[f(a), R]O C f [D(a, rt]. PROOF. Consideration of the translate fr - a] (z) �f f(z - a) and the func tion fr - a] - f(a) shows that the assumption a = f(a) = 0 is admissible. If g(z) �f f(z) - !, (O)z and I z l < r, then g(z) = Jrro,z] [!, (w) - !, (O)] dw, 1 I g(z) l :::; I z l · 1 I!, (zt) - !, (O) I dt. Cauchy ' s formula implies that when I w l < r, !'(u) _ !'(u) ] du = � r f'(u) d , r [ !, (w) - !, (O) = � 27rl J1 l r ( - w) 27rl J1 l r -w U Iwl I w l M ' 27rr = I f, (w) - f, (0 I :::; r - 1-w 1 M, 271' r ( r - I w I ) 2 I g(z )I :::; l z I Jto M r _I z ll zt l t dt = r I_z l l z l · 21 · M. The triangle inequality implies If(z) - f' (O)z l 2': I!, (O) I ' I z l - If (z) l. Be cause � :::; I!, (O) I , (9.2.3) I f(z)l 2': I!, (O) I ' I z l - I g (z)1 2': I!, (O) I ( I z l - r �1 I �Z I ) . Since I z l < r, the maximum value of the first factor in the rightmost member of (9.2.3) is achieved when I z l = l1 � (1 - �) r « r). The maximum value itself is (3 - 2 v2)r and 9.2.1 applies when X = Y = C, V = D(O, l1t, and r5 = ( 3 - 2v2)r I!, (O) I . D 9.2.2 LEMMA.
>
u
=
u
u
=
u u
u
Chapter 9. Defective Functions
372
f
f(O) = 0, 1' (0) = 1, and R �f I l f l � l I oo ,
9.2.4 Exercise.
If E H (U ) ,
9.2.5 Exercise.
a) If E H (U ) , then '1jJ : U '3 z r-+
then f(1U)
:J
D(O, Rt · [Hint: 3 - 2v2 > -61 .]
continuous. b) If
f
I I'(z) 1 ( 1 - I z l ) IS
'1jJ ( z ) cle=f M, and R cl=ef (3-2 In) a E , '1jJ ( a) = max 2 M, z E llJC then f(1U) D(f(a) , Rt . c) R > I f��) 1 � 112 , l --' I a l then [Hint: M = (1 - l a l ) I I' (a) 1 and if I z - a l < 2 I I'(z) 1 :::; 2 11'(a) 1 ; - V
IT 1C \U
:J
9.2.1 applies.]
( 1 )
It is a short step from 9.2.5 to A: b = f(a) and f(1U) :J D b, 1 2 [ 9.2.6 Note. A heuristic (but invalid) argument for A is the following. If 11'(0) 1 � 1, for z near 0, I f(z) - f(O) 1 � 1 ; 1 . Thus, if r is small and positive, each point in the Jordan curve image Jr �f f[ Go(r)] (the image under of the perimeter of the open disc D(O, rt) is at a distance not less than r 2" from 0 [= f(O)]. Thus
f
:J
f(1U) J [D(O, r)O] = U J [Go(s)] O�s 0
10. 1 . 1 1 Exercise. There is no entire function F such that F l n =
I.
g �f exp (�), then g E H (fl) and g(z) 2 = Z in fl. b) There is no entire function G such that G l n = g. c) If h(z) �f Vi = )1 + (z - 1) ... ' 1 = 1 + f (� - ) (� -�? (� - Tn) (z _ 1) Tn �f f (z _ 1) Tn 10. 1.12 Exercise. a) If
Tn =1
n=O
Cm
then r 1 = 1. If D(a, rr c fl, then [g, D(a, rr] is an analytic continuation of [h, D(I, lr]. Despite the conclusions in 10. 1 . 10 and 10. 1. 11, if a �f a + i b E fl and a
I (n) ( ) ( Z _ ) " �-f ""' cn ( Z _ ) " � I( Z ) �-f ""' � (Xl
n=O
, n.
a
(Xl
n=O
a
,
(10.1.13)
then ra = 1 0' 1 ; if c < 0, then D ( a , rat n (-00, 0] i- 0: D ( a , rat is not contained in fl. Consequently, k denoting the function represented by the series in (10.1.13), if z E D ( a , rat and 'S(z) < 0, then k (z) i- I(z), even though [k , D ( a , rat] is an analytic continuation of [h, D(I, lr].
Chapter 10. Riemann Surfaces
384
In D ( a , rat n { z of [h, D(l , 1 rl .
'S( z) < O }
there are two analytic continuations
re-arrangement of series
The process of can (sometimes) yield an analytic continuation [k, D ( a , Rc )Ol of [h, D(l, 1 rl that is different from (I, Q) .
a
f
'
a
10.1.14 Example. If E H (Q), E Q, and J ( ) -j. 0, the Inverse Func tion Theorem, 5.3.41, implies that for some positive in H {J [D ( , rn rl engenders == z. Furthermore, [ , D( there is a such that
r, ar g f I D(a ,r) O g a, r analytic continuations to other function elements. Local inverse functions like g above, the local inverse of exp exp, and sin constitute a particularly g
0
0
rich source of analytic continuations exhibiting the phenomena described above. In fact, 10.1.6 is based on the local inverse of the entire function z r-+ exp(z).
f
a
r
a
regular f,
10.1.15 DEFINITION. WHEN E H (Q) , E 8(Q) , > 0, IS A OF f IFF SOME [ , D( rl IS AN IMMEDIATE ANALYTIC CONTINU ATION OF (I, Q) . WHEN b E 8(Q) AND b IS NOT A REGULAR POINT OF OF b IS A
point
g a, r singular point f.
The set of regular points of f may be empty, e.g., if Q C (in which case the set of singular points is also empty) or if f is such that 8(Q) is the natural boundary for f (in which case the set of singular points is 8(Q).) A singular point a of f is isolated iff f is represented by a Laurent 00 series, L cnz n and for some negative n, Cn -j. O. n==-(X) However, if S �f {O} U { z : mrz = I } and =
f(Z) cle=f sm. ( -1; ) , g( z) �f
{ (� ) csc o
if z tt S otherwise
then {J, g} c H (C \ S) , each point of S \ {O} is a pole of f and an isolated essential singularity of g and 0 is a nonisolated singularity of both f and g. 00
L cn zn IS NONNEGATIVE n=O AND Rc = 1, THEN 1 IS A SINGULAR POINT OF f : 1U '3 z r-+ L cn zn . 10. 1.16 THEOREM. (Pringsheim) I F EACH cn IN
00
n =O
385
Section 10. 1. Analytic Continuation
Tn z �) n PROOF. Since z n = � �O (m) ( 2-n-Tn , the re-arrangement Tn
=
(10. 1.17)
00
of L cn z n diverges if I z l > 1. Owing to the hypothesis Cn 2': 0, n E N+ , n=O Fubini ' s Theorem (4.4.9) implies that if 0 < z < 1, the inversion of the order of summation is valid in (10.1. 17):
� '"z" � [t> (;,) (Z2:_� l = �O Cn [� (:) 2n�Tn 1 (z _ �) Tn �
�
(10. 1 . 18)
� :,�D (z - D If 1 is not a singularity of f , for some positive there is a function element [g , D(I, t l that is an immediate analytic continuation of (I, 1U) . Hence �
!(
�
�
r
r
the distance d between "21 and some point of the boundary of 1U U D(I, r t 1 exceeds r the right member of (10 .1.18) converges for some z in (1, 00 ) , a contradiction. D ' 10.1.19 Exercise. Pringsheim s Theorem (10. 1.16) obtains if the hy pothesis Cn 2': 0, n E N+ , is replaced by � ( cn ) 2': 0, n E N+ , and it is as00 00 sumed that the radii of convergence of both L en z n and L � (en) z n are n=O n=O one.
g ( Tn ) (�) For g : 1U 3 z r-+ L � (cn ) zr> ' L m., z - -I Tn Tn=O n=O diverges if z 1. Furthermore, g ( Tn ) (�) = � [f( Tn ) (�)] and 00
00
[Hint:
>
f(Tn) ( 1 ) � m! "2
(z �) Tn _
( 2)
386
Chapter 10. Riemann Surfaces
diverges if z > 1 ( whether
converges or diverges when z > 1).] 10. 1.20 THEOREM. IF THE RADIUS OF CONVERGENCE =
Re OF
P(a, z) �f L cn (z - a) n n=O IS POSITIVE ( AND FINITE ) , FOR SOME b IN ea (Re), NO FUNCTION ELE MENT [g, D(b, sr] IS AN IMMEDIATE ANALYTIC CONTINUATION OF [P(a, z), D (a, RetJ , i.e., SOME POINT b OF ea (Re) IS A SINGULAR POINT OF I : D (a, Ret r-+ L en (z - a) n . n=O PROOF. Otherwise, each b of ea (Re) lies in an open annular sector 00
and there is a function element (jb, Ab ) that is an immediate analytic con tinuation of [ I, D( a, rr]. Since ea (Re) is compact, ea (Re) contains a £1M nite set { bmL 1, and z Re27rqi , then I Sp k (Z) 1 t 00: n=O overconvergence is absent. In 10.1.24 the sizes of the gaps [the sequences of successive zero coefficients] increase rapidly, while the sizes of the nongaps [the =
k
OF 10.1 .24, IF c i- 0, =
a
a
E
.
a
E
=
00
,
E
00
,
=
=
sequences of successive nonzero coefficients] may increase as well. Hadamard ' s result asserts that if the size of each nongap is one and the sizes of the gaps increase sufficiently rapidly, the boundary of the circle of convergence is a natural boundary. What follows is an illustration of what can happen when the gaps are Hadamard-like per the hypotheses of 10.1.26 and the sizes of the nongaps also increase sufficiently rapidly.
Section 10. 1. Analytic Continuation
If P(z) �f Z ( Z + 1), then
389
00
(10. 1.28)
converges and defines a function f holomorphic in each component of Q �f { z : I P(z) 1 < I }. A point a is on the boundary of a com ponent of Q iff I P(a)J = 1. One component, say C, contains, for some maximal positive r, D(O, rt. Owing to 10.1.26, the bound aries of the components of Q are natural boundaries for the restrictions of f to those components. Since S �f Co( r) n 8(C) "I- 0, if a E S, then a is a singular point of f. In D(O, rt, f is repre sented by a power series P(O, z) calculable from (10. 1.28). Further more a given power of z appears in at most one term of (10.1.28) . Hence both the gaps and the nongaps in P(O, z) are Hadamard like. Overconvergence n occurs for some z in Q \ D(O, r) ("I- 0). 2 If H (z) �f L �, n the radius of convergence of the right member n= l is 1. Owing to 10.1. 26, '][' is a natural boundary for h. On the other hand, whereas I f I and I g l in 10. 1.23 are unbounded in 1U, 7[2 if I z l < 1, I H( z)1 ::::: L n12 6 · ex:>
ex:>
=
n= l 1 � z n and the radius of conver If I z l < 1, then f(z) �f _ � z -_1 = n=O gence of the right member is 1, while 1 + iO is the unique singular point of f. Nevertheless, if I z l > 1 and { s nk (z)} kE]\/ is a sequence of partial sums of L z n , {S nk (z)} kE]\/ diverges: overconvergence n=O is absent. 00
There is an extended discussion of the phenomena noted above in
[Di].]
For the functions f resp. g in 10.1.6 resp. 10.1.12, if (II , Q) resp. (gl , Q) is an analytic continuation of [J,z D(I, 1 t] resp. 2[g, D(I, It] ' the Identity Theorem implies that in Q, eh ( ) = z resp. g l (z) = z. The three results that follow elaborate on this theme. 10. 1.29 Exercise. For the polynomial
Tn l = l ,
...
,Tn n = l
a
'TTl l , · · · , 'TTl n
n n,
Tn 1 . . . Tn w1 w
Chapter 10. Riemann Surfaces
390
if { (II , Q) , . . . , Un , Q)} is a set of function elements such that on Q,
P (II , · · · , fn) = O, for any set {(g l , Q I ) , . . . , (gn , QI ) } of function elements that are analytic continuations of the { (II , Q) , . . . , Un , Q) } , the equation is valid on QI . [Hint: The Identity Theorem for holomorphic functions applies.] 10.1.30 Exercise. If, in the context of 10.1. 29, f is a solution on Q of the differential equation P (y, y' , y" , . . . ) = 0 and (g, QI ) is an analytic continuation of (I, Q) , on QI , P ( g, g' , g" , . . . , ) = O. 10.1.31 Exercise. The conclusion in 10.1. 29 remains valid if P is rea'm l, 'm n W � l • • • w;:' n that converges placed by a power series L CXJ , • • • ,CXl
'TTl l = l , ·· · , 'TTln =l in a nonempty polydis c X:= I D (0 , r ) 0 [ 10.1.32 Remark. The phenomena in 10. 1.29-10.1.31 are ex amples of the Permanence of Functional Equations (under ana •••.
k
•
lytic continuation).
Thus, although analytic continuation permits the creation of new function elements from old, when 1 � m � n, (I'm , Q) resp. (grn, Q) are analytic continuations of one another, and
(in this instance, the second members of all the fUllction elements are taken to be the same) , nevertheless, functional relations among the f'm 1 � m � n, persist among the grn , 1 � m � n .] '
w i - w � , then (II , Q I ) � ( z r-+ z, q , (12, Q2 ) �f ( z r-+ -z, q are function elements such that P (II , II ) = P ( 12 , h) = O. Yet neither of (II , Q I ) and (12, Q2 ) is an analytic continuation of the other. The converse of the principle of the Permanence of Functional Equations (under analytic continuation) is false.
10.1.33 Exercise. In 10.1.29, if
n = 2 and P (W I , W2 ) �f
391
Section 10.2. Manifolds and Riemann Surfaces 10.2. Manifolds and Riemann Surfaces
The developments that follow organize the study of functions such as In : D ( 1, 1 t '3 z r-+ In z E C, vrD(I, It '3 z r-+ Vi E C,
and other functions afflicted with ambiguity ( multivaluedness) when their original domains are extended, e.g., to C \ {a}. As the discussion in Section 10.1 reveals, a power series P( a, z ) can engender analytic continuations to power series PI (b, z ) and P2 (b, z ) such that PI (b, z ) -j. P2 (b, z) . The definition of the word function as it is used in mathematics makes the term multivalued function an oxymoron, although there is a temptation to describe as multivalued a function f that is locally represented by P( a, z), PI (b, z), and P2 (b, z). Further discussion can be conducted systematically in the context es tablished by the following items. A complete analytic function is a collection CAF �f {(Iv, flv)} v EN of function elements such that: a) each is an analytic continuation of every other; b) any function element that is an analytic continuation of a function element in CAF is (also) in CAF: CAF is a maximal set of function elements each of which is an analytic continuation of any other. The function elements (II" ' fll" ) and (Iv, flv) in CAF are a-equivalent, i.e. , UI" ' fll") Uv , flv), iff: a) a E fll" n flv; b) in some neighborhood N(a), fl" I ( a) = fv I ( ) " Thus to each a in the union fl �f U flv v EN there corresponds a set of "'a-equivalence classes of function elements (II" ' flv) for which a E flw When (I, fl) E CAF and a E fl, the "'a-equivalence class to which U, fl) belongs is the germ or a branch of (I, fl) at a, v. 10.2.1, and is denoted [I, a]. As the discussions of 10. 1 .10 and 10.1.12 reveal, it is quite possible for a function element (g , (for which a E to belong to CAF while, in the current notation, [I, a] -j. [g, a]. A result proved below and due to Poincare, implies that the cardinality of all germs at a cannot exceed No . The set W �f { [I, a] : a E fl, (I, fl) E CAF } of germs is, in recogni tion of its originator, Weierstrafi, the W-structure determined by any (hence every) function element in CAF. (A topology for W is given in 10. 2.4 below.) Associated with W are the projection p : W '3 [I, a] r-+ a E fl and the map f : W '3 [f, a] r-+ f(a) E C. •
•
N N "'
a
a
•
n)
•
•
n)
Chapter 10. Riemann Surfaces
392
10.2.1 Exercise. a) When a E C and P(a, z ) is a power series with a positive radius of convergence Ra, then P( a, z ) represents some function f and an associated germ [I, a]. b) For a fixed, the correspondence
{ P(a, z)
: Ra > O } -+
{ [f, a] : a E fl, f E H (fl) }
between the set Sa �f {(P( a, z ) , Ra) } of all pairs consisting of a power series P( a, z) converging at and near a and the associated radius of convergence Ra and the set Hf, an of all germs at a is bijective. c) If UI" ' fll" ) and Uy , fly ) need not be "-'b-equivalent. Nevertheless, UIL ' fllL ) and Uy , fly ) are analytic continuations of one another. d) If a -j. b two of the equations [f, a] = [g, b], [I, a] = [g, a], [I, a] = [I, b], are meaningless. For any a in fl, a function element U, fl) and one of its analytic continuations such that a E fl n the numbers f( a ) and a) may differ, v. 10.1.10 and 10. 1.12. Nevertheless, they are regarded as values of the multivalued function that arises from analytic continuation. On the other hand, f is a true (single-valued) function on W and the range or image f(W) accurately reflects the different values f(a), g(a), . . .
(g, n)
n,
g(
,
10.2.2 Exercise. The set fl '!gf ex:>
U fly is a region. What is fl in 10.1.21?
"'EN
[Hint: The series L .;. arises by integrations and algebraic trans n=! n formations applied to L zn .] n
00
n=O
10.2.3 Exercise. The function f is well-defined on W. The description of CAF suggests a topology derivable by pasting to gether the fly used to provide analytic continuations. However, the accu
rate description of such an informal topology is beset with the complications arising from the presence in CAF of equivalent function elements. A topol ogy is more readily attached to the set W of equivalence classes, i.e., germs or branches, according to 10.2.4 DEFINITION. WHEN V IS AN OPEN SUBSET OF fl AND
[I, V] �f { [f, a] : a E V }.
f E H (V) ,
Section 10.2. Manifolds and Riemann Surfaces
393
10.2.5 Exercise. The set
T �f { [I, V]
: V an open subset of Q, for some Q,
( j, Q)
E CAr }
is a Hausdorff topology for W. 10.2.6 THEOREM. WITH RESPECT TO T, p IS CONTINUOUS AND OPEN.
PROOF. If a E V E 0(((:) and [I, a] E p-l (V) , then [I, V ] is a neighborhood of [I, a] and p( [J, V] ) = V: p is continuous. For any open subset V of Q, p([I, V] ) = V, whence p is open. D 10.2.7 Exercise. If V is an open subset of Q, then p l [J, v] is injective. (Hence p is locally a homeomorphism.) [ 10.2.8 Note. For a in Q, a in C, and germs [I, a] , [g, a] , the germs [aJ, a] , [I + g, a] , and [lg, a] are well-defined. Thus, for each a in Q there is the C-algebra !i a
�f { [I, a]
: J holomorphic in some N
(a) }
of germs of functions holomorphic at and near a. Generally, for a category C, e.g. , the category of C-algebras, and a topological space X, a sheaJ S [Bre] is defined by associating to each U in O(X) an object S(U) in C. It is assumed that: When { U, V } C O(X) and U C V, there is a morphism •
Pb : S(V) r-+ S( U ) •
• •
When { U, V, W } c O(X) and U e V e V w. Pw u = pu Pv ,
.
W,
then
0
p� is the identity morphism. When {U>.} ), EA C O(X), U �f U U)" k l ' k2 in S( U ) are the ), EA same iff for each A, pg>. (k I ) pg>. (k2 ) .
For U), and U above, if
=
k), E S (U), ) , A E A, { W �f U), n Ul" -=j:. (/)} ::::} {p� (k),) = p� (kl" ) } , there is in S( U ) a k such that for every >., pg>. (k) = k),.
394
Chapter 10. Riemann Surfaces
When x E X, the filter V �f {V(x) } of open neighborhoods of x is partially ordered by inclusion: U(x) -< V(x) iff U(x) C V(x). The stalk Sx at x is the set of all {kv } v E V such that For s E sx , if p(s) �f x, then p maps S �f
U
Sx
onto X. The
set {p - l (U) : U E O(X) } is a topology T for S; p is open and T is the weakest topology with respect to which p is locally a homeomorphism. If X is a Hausdorff space, so is S. When the objects in the category C are algebraic, e.g. , when C is the category of groups and homomorphisms or the category of C-algebras and C-homomorphisms, as x varies in X the results of the algebraic operations within the stalk Sx are assumed to depend continuously on x. For example, when X = C and C is the category of all C-algebras, for each open subset V of C, the set S(V) may be taken the set of all functions holomorphic in V, and when U C V, pi; maps each f in S(V) into f l u ' Then: S is the sheaf of germs of holomorphic functions. For a in C, Sa is the stalk at a and consists of all germs [I, a] . The map p : S r-+ C is that given iri the context of W : p maps each germ [I, a] onto its second component a. A W-structure W is a connected subset of S. The elements [oJ, a] , [I + g, a] and [ lg , a], maps from C to S are continuous.] xE X
as
•
•
•
•
•
as
10.2.9 Exercise. The topology induced on each stalk of a sheaf is discrete. (Hence, when the stalk is an obj�ct in a category of topological objects, the (discrete) topology induced by T on the stalk need not be the same the topology of the stalk viewed an object in its category.) as
as
Q
10.2.10 THEOREM. (Poincare) IF a E THE CARDINALITY OF THE SET OF GERMS [I, a] AT a DOES NOT EXCEED No . PROOF. Each germ [I, a] corresponds to some function element (such that a E If [ , a] corresponds to (again a E then is an analytic continuation of (I, Thus there is a finite chain
Qv). g
in which
Qv).
(g, QJL)
QJL) '
(f, Qv) (g, QJL)
(ik, QVk ) is an immediate analytic continuation of (ik-l , Qvk _ l ) ' QVk contains a point p + iq for which {p, q} C Qi, i.e.,
2 :::; k :::; m. Each
Section 10.2. Manifolds and Riemann Surfaces
395
p + iq is a complex rational point. Thus, corresponding to each finite chain
C, there is a finite sequence of complex rational points. The cardinality of the set of all finite subsets of ((f is No · D Thus, if Z E Q and Sz �f p - 1 ( z ) (the stalk over z ) , #(sz) :s; No. If r > 0 and V is an open subset of Q, each [I, z] (E sz ) is in a neighborhood [I, V] . Furthermore, p : [I, V] r-+ V is a homeomorphism. Consequently, W may be viewed as consisting of sheets lying above Q and locally homeomorphic to Q. If a E Q, for some positive r, V �f D(a, rt lies under a set of homeomorphic copies of V, One copy in each of the sheets. In 10.2.11 and 10.2.12 there are precise formulations of the preceding remarks. 10.2.11 Exercise. As z ranges over Q, the cardinality #(sz) remams constant. [Hint: If #(sz) �f k E N, then #(sz) = k near z. If
analytic continuation from z to w yields a contradiction.] 10.2.12 Exercise. If 1 E E, #(sz ) == 1 for the associated W-structure W. 10.2.13 Exercise. If a E Q, for some N(a), p - 1 [N(a)] consists of (at most count ably many) pairwise disjoint homeomorphic copies of N(a). The following variant of the ideas above leads to greater flexibility of the discussion. The W-structures introduced thus far are extended to analytic structures that include so-called irregular points [Wey] . 00 The map I : Q 3 z r-+ L cn (z a ) n may be viewed as an analytic
n=O description of a complex curve
{
-
(z, w )
: w =
� cn (z - a)n }
in Q x C.
The same object may be described alternatively by the pair z =
00
a + t, W = Co + L1 cntn n=
(10.2. 14)
of parametric equations. Extended somewhat further, the parametric equa tions (10.2.14) are replaced by a pair 00
00
P(t) �f L an t" , Q(t) �f L (3rn t'n n =k
for which the following conditions are imposed.
Chapter 10.
396
Riemann
Surfaces
a) { k , l} C Z ; b) P and Q converge in some nonempty punctured disc .
D(O,
(if
c)
rr clef= { t
:
0
< ItI < r }
k and l are nonnegative, P and Q converge when t = 0, i.e., r
D (O , n ;
m
{ { {t l , t2 } C D (O, rr } 1\ {P (t I ) = P (t2 )} 1\ {Q (t I ) = Q (t2 )} }
::::} {t l = t2 } '
When k or l is negative, 0 is the only pole in D(O, rr of the corre sponding P or Q. Hence, if, e.g., P ( O) = 00 = P (t 2 ), automatically t 2 = O. Thus there is a uniquely defined injection
rr 3 t r-+ [P(t ) , Q(t )] E C� . The parameter t is a local uniJormizer. The Junction pair (P, Q) , subjected L :
D(O,
to a)-c) , is now the object of interest. Various parametric representations can represent the same curve, e.g.,
{ (cos t, sin t) : -7r < t < 7r } and { ( 11 +- 77: , � ) 1+7
: -00
< 7 < 00 }
are different parametric representations of
2 y2 = 1, x
{ (x, y) : x +
>
-1 } .
Function pairs [P (t ) , Q (t ) ] and [R(7), S(7)] (both subjected to a)-c)) are regarded as equivalent and one writes [P (t), Q( t ) ] rv [R( 7), S ( 7)] iff the local uniformizer 7 is representable in the form 00
7
= L "Intn , "11 :=J 0, n= l
(10.2.15)
and the series (10.2.15) converges in some open disc D(O, rr. 10.2.16 Exercise. The relation rv described above is an equivalence relation. 3
The rv-equivalence class containing the function pair (P, Q) is denoted (P , Q) or simply 3· The set of all 3 is 3 and the elements 3 of 3 are points.
Section 10.2. Manifolds and Riemann Surfaces
397
10.2.17 Exercise. For the maps
3(P, Q) r-+ P(O) E ((:(3 : 3 3 3(P, Q) r-+ Q(O) E ((:, ( (3) : 3 3 3(P, Q) r-+ [P(O), Q(O)] E ((:2 , the complex numbers [3 (P, Q)] and (3 [3 (P, Q)] are independent of the representative (P, Q). Thus the notations 0' (3) and (3(3) unambiguously define complex num bers: and (3 are in ((:3 . a :
33
a,
a
a
10.2.18 DEFINITION. THE TOPOLOGY T OF 3 IS THE WEAKEST TOPOL OGY WITH RESPECT TO WHICH : 3 r-+ ((:2 DEFINED ABOVE IS CONTINU OUS. 10.2.19 Exercise. If E 3 and is a nOnempty open disc L
3(P, Q)
D(O, rt
on which the map : D(O, rt 3 t r-+ [P(t), Q(t)] E ((:2 is injective, a typical neighborhood N [3 (P, Q)] is (D(O, rn and consists of all 3 (15, described as follows. For some to in D(O, r t and all t such that to + t E D(O, r t, L
L
Q)
N [3 (P, Q)] consists, for all im mediate analytic continuations (15, Q) by rearrangement of the pair (P, Q), of the points 3 (15, Q) . 10.2.21 Exercise. If (P, Q) (R, S), each neighborhood N[3 (P, Q)] contains a neighborhood N[3 (R, S)]. The base of neighborhoods at a point 3 in 3 may be defined without 10.2.20 Exercise. By abuse of language,
rv
regard to the particular parametrization.
10.2.22 Exercise. As defined by neighborhoods described above, T is a
Hausdorff topology for 3. 10.2.23 Exercise. Each point 3 in 3 is contained in a neighborhood N(3) homeomorphic to 1U. 10.2.24 Exercise. The maps a and (3 are local homeomorphisms with respect to the topology T.
[� cn (z
rt1 is a function element, [I, a] the corresponding equivalence class, and (P, Q) is a function pair
10.2.25 Exercise.
If
- a) n , D ( a ,
398
Chapter 10. Riemann
created by unijormization, i.e., P(t) = a + t,
Q(t) = L cntn , the map
I : W 3 [I,
is a
(1", T) homeomorphism.
a] r-+ 3 (P , Q) E 3
00
Surfaces
n=O
In many discussions of sets of equivalence classes, e.g., in LP ( X, J1 ) , the distinction between an equivalence class and one of its representatives is blurred. Thus, when ambiguity is unlikely, no distinction is made between an equivalence class 3 and one of its function pairs ( P , Q) . 10.2.26 Example. The following are some important examples of function pairs. P (t) = a + t,
P (t) = C k , k E N,
00
Q(t) = L bm tm , 00
m=O 00
m=l
Q(t) = L bm tm , l E Z. m=l
(10.2.27) (10.2.28 ) (10.2.29)
The function pair in: (10.2.27) corresponds to a function that is locally invertible near a; (10.2.28) corresponds to a function that conforms to the behavior con sidered in 5.3.47; (10.2.29) corresponds to a function with a pole.
10.2.30 Example. By appropriate reparametrizations, any function pair can be represented in one of the forms (10.2.27), (10.2.28), or (10.2.29). (10.2.27). If the original pair is antn , bm tm and a l -j. 0, a reparametrization is
(� %:;0 )
(10.2.28). If k 1, ak -j. 0, and the original pair is (ao + aktk + . . . , t, bmtm) , >
Section 10.2. Manifolds and Riemann Surfaces
399
for a Ck such that c� = a k , a reparametrization is
(10.2.29). If k > 0 and the original pair is a- k :=J 0, and c� = a_ k , a reparametrization is ", cptp t -_ � '" O'qTq , 0' 1 _- -C11 , T �-f p� q= 1 =1 ex:>
ex:>
Those points 3 representable in form (10.2.27) are the regular points; those representable in forms ( 10.2.28 ) and (10.2.29) are the irregular points. More particularly, points representable in form (10.2.28) are branch points of order k; those representable in form (10.2.29) are poles of order k. When k > 1 in form ( 10.2.29), the pole is branched.
analytic structure
10.2.31 DEFINITION. A SUBSET AS of 3 IS AN IFF # (AS) > AND: a) ANY TWO POINTS OF AS ARE THE ENDPOINTS OF A CURVE "/ 3 r-+ E 3 ; b ) WHEN 3 E 3 AND FOR SOME CURVE E AS AND = 3 , THEN 3 E AS. Each W is a subset of some unique AS(W): W e AS(W) . If each point of AS(W) is a regular point, W = AS(W) . By abuse of language, an analytic structure AS is a 3.
,,/(0)
:
1 [0, 1] t "/(t) ,,/(1)
,,/,
curve-component of
10.2.32 Example. Under the convention whereby � is identified with 00,
(
�)
o
i.e., when the discussion is conducted in Coo , One neighborhood of the function pair :F �f P ( t) = t, Q(t) = corresponds to { t : t E C, I t I > O } , a neighborhood of 00, v. Section 5.6. The analytic structure engendered by analytic continuation is Coo .
Chapter 10. Riemann Surfaces
400
Owing to the generality with which function elements are defined, the analytic structure AS is larger than the analytic structure W presented earlier. The inclusion of irregular function elements permits the adjunction to W of branch points, poles, etc. Henceforth, Weyl's adaptation [Wey] of Weierstrafi ' s ideas for analytic structures is invoked where it is helpful. 10.2.33 LEMMA. EACH IRREGULAR poINT 30 LIES IN A NEIGHBORHOOD
N (30 ) IN WHICH ALL OTHER POINTS ARE REGULAR.
PROOF. If (P, Q) E 30, for some positive
r, if ° < l a l < r, then
and p' I v ¥= 0, since otherwise, the Identity Theorem implies p' 0, a con tradiction. The neighborhood N (30 ) corresponding to D(O, rt meets the requirements. D ==
10.2 .34 THEOREM. AT MOST COUNTABLY MANY POINTS IN AN ANALYTIC STRUCTURE ARE IRREGULAR.
PROOF. From 10.2.33 it follows that each 3 is contained in a neighborhood
N (30)
consisting (except possibly for 30 itself) of regular points. One of these regular points corresponds via the local homeomorphism to a p + iq in QI + iQl. Thus the irregular points are in bijective correspondence with a subset of QI + iQl (cf. PROOF of 10.2.10). D 10.2.35 Example. If clef
Cn =
(�) (� ) . . . [-(2�- I) ] n!
00
and I
z - 1 1 < 1,
L cn (z - I) n converges and represents a function fo such that for n=O Z in D(I, It, [fo(zW = z. The two parametric representations of Vz are [R±(t),S±(t)] � (t + 1, ± cntn ) there are two square roots of z.
then
�
:
2zt
z
More generally, if :::; k E N, there is a sequence { cn } n E]\/+ ' found by formally differentiating and evaluating the results when = 1, and for 00 in D(I, It, L cn( z - It converges and represents a function fo(z) such
z
z
n=O
that for in D(I, It,
[Jo(zW
=
z: Parametrizations of (fo, D(I, 1) ° ) are
Section 10.2. Manifolds and Riemann Surfaces
[Rq(t), Sq (t)] �f (t + 1, (different) kth roots of
401
� ( � ) cn t n )
z.
exp 2 7ri
, 0 :::; q :::; k
- 1: there are k
Via 10.2. 19, there are k pairwise disjoint neighborhoods
each homeomorphic to 1U. The corresponding AS provides a k-fold cover of C, v. Section 10.3. If 0 :::; q :::; k 1, "I : [0, 1] 3 s r-+ "I(s) E 3 is a curve, and for s in [0, 0.5), "I(s) (Ps , Qs ) is regular, v. (10.2.27) , while "1 (0 . 5 ) = (Rq , Sq ), there are k possible continuations of "1 ( 0 . 5) along the rest "I ([0 .5, 1]) of the curve. If "I : [0 , 1] 3 t r-+ "I(t) E 3 is a curve and, when
�f
-
:::; t :::; to < 1, each "I(t) is a regular point but "I (to) is a branch point of order k, according to which of the k different choices of representation defines "I (to), the curve "I is One of k curves "11 , . . . ,"Ik , say "Ii such that if 0 :::; t < to if to :::; t :::; t 1 :::; 1 . The phenomenon just described is the genesis of the term branch point and the particular "Ij is a branch. That "Ij may lead to another branch point for some t l in (to, 1). One of the branches corresponding to a t l in (to, 1) can be one of the "Ii discarded at to · 10.2.36 Exercise. In the context above, if 5 is the set of t such that "I(t) is an irregular point, #(5) E N+ . [Hint: The argument in the PROOF of 10.2.33 applies.] o
The further study of analytic structures is facilitated by the next dis cussion of related topological questions. 10.2.37 DEFINITION. A TOPOLOGICAL SPACE X IS: a) IFF ANY TWO POINTS OF X ARE THE ENDPOINTS "1 0 AND "1 1 OF A CURVE r-+ X; b) IFF FOR EACH x IN X, EACH NEIGHBORHOOD N(x) CONTAINS A NEIGHBORHOOD V(x) SUCH THAT ANY TWO POINTS IN V(x) ARE END POINTS OF A CURVE SUCH THAT "1 * C N(x) ; c) IFF X IS CURVE-CONNECTED AND EACH LOOP
curve-connected () () locally curve-connected
simply connected
"I : [0, 1]
"I
"I : [0, 1] r-+ X
Chapter 10. Riemann Surfaces
402
d)
IS LOOP HOMOTOPIC IN X TO A CONSTANT MAP;
locally simply connected IFF X IS LOCALLY CURVE-CONNECTED AND FOR EACH x IN X, EACH NEIGHBORHOOD N(x) CONTAINS A NEIGH BORHOOD V(X) SUCH THAT EACH LOOP "( FOR WHICH "((0) x ( THE LOOP STARTS AT x), "((I) x ( THE LOOP ENDS AT x), AND "( * C V(X) ( THE LOOP IS CONTAINED IN V(X)) IS LOOP HOMOTOPIC ( VIA SOME CONTINUOUS MAP F OF [0, 1] 2 ) IN N(x) TO THE CONSTANT MAP 15 : [0, 1] 3 t r-+ r5(t) x. ( IN THE NOTATION OF 1.4.1, =
=
=
"( "' F,N(x)
15.)
curve-component.
MAXIMAL CURVE-CONNECTED SUBSET OF X IS A 10.2.38 Example. The punctured open disc 10 is curve-connected, locally curve-connected, locally simply connected, but simply connected. The union 1Ul:JD(2, is locally curve-connected, simply connected, locally simply connected, and connected. The A
•
•
•
not 1r not topologist's sine curve { (x, y) : y sin (�) , ° < x � 1 } l:J { (0, y) : - 1 � y � 1 } ' in the topology inherited from ffi.2 is connected, not curve-connected, and not locally connected ( no point on { (O, y) : - 1 � y � I } lies in =
a connected neighborhood ) .
n chart at
10.2.39 DEFINITION. A CONNECTED HAUSDORFF SPACE X IS AN IFF: FOR EACH POINT IN X THERE IS A PAIR (A CONSISTING OF A NEIGHBORHOOD OF AND, FOR SOME NONEMPTY OPEN SUBSET U OF THERE IS A HOMEOMORPHISM
dimensional cumplex manifold x x) •
•
{N(x), ¢ } N(x) x en , ¢ : N(x) 3 x r-+ U. FOR CHARTS {N(x), ¢} AND {N(y), 1jJ } SUCH THAT N(x) n N(y) -j. (/) THERE IS THE transition map t<jJ¢ �f 1jJ ¢-l : ¢[N(x)] n 1jJ (N(y) r-+ ¢[N(x)] n 1jJ (N(y). WHEN EACH t<jJ¢ IS IN H (W) , i.e., WHEN a �f (a l , . . . , a n ) E W, AND THERE ARE n POWER SERIES c( k ) Tn l .... ,Tn n (Zl - a l ) Tn l . . ( n - an ) Tn n 1 _< k _< n 0
, z
Section 10.2. Manifolds and Riemann Surfaces
403
t7jJ¢ ON EVERY COMPACT SUBSET OF (a , r t CONTAINED IN W, X IS AN n -dimensional complex analytic manii fiold. THE SET A � {{N(x), ¢} L E x IS AN atlas. Two charts {N1 (x), ¢d and {N2 (x) ' ¢2 } are holomorphically compat ible iff the map ¢2 ¢� 1 : ¢ 1 [N1 (x) n N2 (X)] r-+ ¢2 [N1 (x) n N2 (X)] is bi holomorphic. CONVERGING UNIFORMLY TO A NONEMPTY POLYDISC X�= l D
0
Two atlases � and � for a complex analytic manifold X are ana lytically equivalent 1(�1 rv �22 ) iff for each x in X, every chart {N1 (x), ¢ 1 } in � 1 is holomorphically compatible with every chart {N2 (x), ¢2 } in �2 . An atlas � is complete iff any chart holomorphically compatible with some chart in � is in �: � is saturated. [ 10.2.40 Note. The assumption that X is a Hausdorff space is not redundant [B e]. For k in N+ U {oo}, there are Ck -compatible charts, and C k _ equivalent atlases. c ) The map t7jJ ¢ may be regarded as a ffi.2 n_ valued function on ffi.2 n . clef . If Zj = X2j - 1 + �X j and t<jJ¢ clef= (U 1 , U2 , · · · , U2n- 1 , U2n ) the 2 sign of the determinant of the Jacobian matrix J [ 8(X 8 (U 1 , . . . , U2n) ] det ( J) �f det 1 , . . . , X2n ) may vary from point to point of X. When det ( J) 0 everywhere , X is oriented by the atlas and X is orientable. When there is no atlas that orients X, X is nonorientable. The Riemann sphere L 2 is an orientable ( I-dimensional ) complex manifold. If 0 < a < b < 1, f z a } , V �f L2 n { (x, y, z) z < b } , u � L 2 n { (x, y, z) then U U V = L 2 . Under the convention � = 0, 1 8 P P U 3 r-+ 8 ( P ) and (3 : V 3 r-+ ( P ) provide charts {U, a} and {V, (3} that constitute an atlas for L 2 . The Mobius strip M S is the set S �f [0, 1] (0, 1) reduced modulo the equivalence relation rv defined by if a -j. O (a, b) rv (c, d) {:} { {a{a == O}c} /\/\ {b{c == dI}} /\ {b = 1 - d} otherwise · >
:
>
00
a :
x
404
Chapter 10. Riemann Surfaces
More intuitively, the Mobius strip is S with O x (0, 1) and 1 x (0, 1) identified according to the rule (0, == (1, 1 For the maps
x) - x). ¢ : [0 , 0.6] x (0, 1 ) 3 (x, y) r-+ (x, y), 'ljJ : [0 . 4, 1] x (0, 1 ) 3 (x, y) r-+ (1 - x, 1 - y), an atlas A for M S is the pair of charts {[O , 0.6] x ( 0, 1 ) , ¢} and {[0. 4, 1] x (0, 1 ) , 'ljJ}. The map t<jJ viewed a map of as
W �f (0.4, 0.6) x ( 0, 1 )
+ + i(l into itself takes the form = The deter minant of the corresponding Jacobian matrix is - 1: M S is not oriented by
t<jJ(x iy) x
- y).
A.]
A2
10.2.41 Exercise. a) If Al and are equivalent atlases for a manifold, the determinants of the corresponding Jacobian matrices are of the same sign. ( Hence the Mobius strip is not orient able. b) If A l and are equiv alent atlases for an oriented manifold, are associated and charts such that the determinant of the Jacobian matrix for n -j. 0, . cle . l t he transItIOn map , 2 =f ' 0f constant sIgn . IS
{ Nl , ¢1 }
)
A2
{N2 , ¢2 } N N l 2 . t '/-' 1 0 ¢2 [Hint: b) For charts { N1 , ¢ 1 } resp. { N2 , ¢2 } in Al resp. A2 such that Nl n Nl n N2 n N2 -j. 0 , a calculation of the determinant of - the Jacobian matrix for ¢ 1 0 ¢:; 1 0 ¢ 1 0 ¢2 - 1 applies.] The set of charts endows X with a topology and with the capacity to support a notion of holomorphy for certain maps defined on X: X is an (n-dimensional) complex analytic manifold. When X and Y are n-dimensional complex analytic manifolds, a func tion F in Y is holomorphic near x iff for some chart { N (x), ¢} and some chart { N[F(x)], 'ljJ}, 'ljJ o F o ¢ - 1 is holomorphic on ¢[N(x)]. If F is holo morphic near each x, F is analytic. For the special case Y = Coo , F is X -holomorph'ic iff F is analytic and F(X) c C; F is X -meromorphic iff F is analytic and F(X) c Coo . When E CX , is harmonic, subharmonic, superharmonic resp. har monic, subharmonic, superharmonic near x iff for each chart resp. some chart { N ( x ), ¢ } , 11 0 ¢- 1 E Ha{¢[N(x)]}, 11 0 ¢- 1 E SH{¢[N(x)]}, - 11 0 ¢- 1 E SH{¢[N(x)]}. --/"
-
x
1l
11
The use of charts permits many discussions of complex analytic man ifolds and functions on them to be carried out locally. In particular, the
Section 10.2. Manifolds and Riemann Surfaces
405
immediate analytic continuation
analytic continuation
notions and are for mulable for appropriate function elements defined for n-dimensional com plex analytic manifolds. Remark. A analytic structure AS (determined by some function element (I, Q)) is a special kind of When E AS, is a neighbor hood of r-+ AS is the associated map, and D(O, is a chart.
[ 10.2.42 lytic manifold. 3(P, Q), {N[3 (P, Q)], - I } L
:
3(P, Q) rr
L
N[3 (P, Q)]
complex ana
Although the study of complex analytic manifolds is a large and important field of mathematics, only the definition of a complex analytic manifold is offered here. Useful references are [BiC, Nar, SiT,
St].]
10.2.43
Riemann surface
DEFINITION. A IS A I-DIMENSIONAL COMPLEX ANALYTIC MANIFOLD. In what follows, most of the discussion will deal with Riemann surfaces. The W-structures W and analytic structures AS have served the purpose of motivating the treatment. Exercise. If R is a Riemann surface determined by an atlas A, there is a holomorphically compatible atlas Al such that each chart o. (The atlas of Al is i.e., 1U and Al is a canonical atlas.) applies.] Riemann's Mapping Theorem
10.2.44 {N(x), ¢} canonical, ¢[N(x)] ¢(x) 8.1.1 [Hint: 10.2.45 Exercise. a) If W is a W-structure and [J, V] E T, then {[J, V],p} =
=
is a chart. With respect to the set of all such charts, the W-structure determined by the function element (I, Q) is a Riemann surface. Exercise. A Riemann surface is orientable. A transition function is a (vector) function
10.2.46 [Hint:
t'j;.p
(lie t2 The Jacobian matrix .J for t<jJ.p is the
( Ux ) uy
vx
Vy
.
(�)
2 2 matrix x
m
8.1.6-8.1.7.] 10.2.47 Exercise. For each convergent series L cn (z - a) n and its circle 'n=O of convergence D( a, r r, the formul32 f(z) �f L cn (z - a) n , P(t) �f + t, and Q(t) �f L cn tn nO 71=0 The Cauchy-Riemann equations apply, v.
=
=
a
00
=
406
Chapter 10. Riemann
Surfaces
[I, a
establish a correspondence between the germ ] and the equivalence class 3(P, Q). Then I : W 3 ] r-+ 3(P, Q) E 3 is an injective and holomorphic map of the Riemann surface W into the Riemann surface AS, the Riemann surface determined by 3(P, Q) (and contained in 3). 10.2.48 Example. The open unit disc 1U is a Riemann surface. If E H (1U) and 8(1U) is a then 1U is the analytic structure de termined by the function element (I, 1U) . Hence , if Q) is an immediate analytic continuation of (I, 1U) , then Q c 1U. Thus, if E Q, any neighbor hood of the germ ] is a subset of the neighborhood 1U] . 10.2.49 Example. The complex plane C is a Riemann surface. If is entire, C is the analytic structure determined by the function element q . One atlas consists of the single chart {C, id }. 10.2.50 Example. The set Coo viewed as the one-point compactification of C, is a Riemann surface. An atlas for Coo is given in 10.2.40. The function element (id , Coo ) determines Coo . The function id is holomorphic in the sense that it is analytic on Coo and id (Coo ) = Coo . Liouville's Theorem (5.3.29) implies that H (Coo ) consists entirely of constant functions. In 10.3.21 and 10.3.23 there are precise formulations of the U niformization Theorem, a significant generalization for simply connected Riemann surfaces of 8.1.1. Two of the closely related consequences, 10.3.24 and 10.3.26 are discussed as well. In Sections 10.5 and 10.6 there are sketches of the proofs of the U niformization Theorem.
[I, a
natural boundary for f, [g, a
f
(g , a
[I,
f (f,
10.2.51 Exercise. The Riemann surface R corresponding to 10.1.6 is homeomorphic to C. If Sn �f { z : z = < < 00, -mr < e � mr and S �f { awl + then R is homeomorphic : (a, E to: a) U Sn resp. b) S.]
[Hint:
(3W2
reiO , 0 r (3) [0, 1) 2 },
}
nEZ
10.2.52 Exercise. A Riemann surface is locally compact and curve connected, locally curve-connected, and locally simply connected. For each chart {N ( x ) , ¢ }, ¢ is a homeomorphism.]
[Hint:
10.2.53 DEFINITION. FOR AN ANALYTIC STRUCTURE AS DETERMINED BY A FUNCTION ELEMENT (I, Q) , AN OPEN SUBSET OF U OF AS , SUCH THAT P cle=f P u IS INJECTIVE IS A
patch.
I
10.2.54 Exercise. For a region 11 of AS, a function F in CQ is
phic on 11, i.e., F E H ( 11 ) , iff for each patch U contained in 11, F (p- l ) E H [P(U)]. 0
holomor
Section 10.2. Manifolds and Riemann Surfaces
407
10.2.55 THEOREM. IF AS IS DETERMINED BY THE FUNCTION ELEMENT (I, Q) THE CORRESPONDING FUNCTION f IS HOLOMORPHIC ON AS.
, a]
=
PROOF. If U is a patch, some [ I is in U and p is injective on U. Fur thermore p(U) �f V is an open subset of and f p - l I l v . D There is a parallel between the study of functions 1 holomorphic in a region Q of and functions F AS r-+ holomorphic in a region 11 on the analytic structure AS associated to I . A local uniformizer t maps a neighbor hood N ( ) into 10.2.56 Exercise. A function F : AS r-+ is holomorphic in a region 11 iff for each 30 in 11 there is a local uniformizer t such that in some neighborhood of 30 , F is representable by a convergent power series:
C
C
:
0
C
C
C.
a
00
C,
n=O
z Iz C - z g z 2 2- z C g(z).
[ 10. 2.57 Remark. If Q c a function I : Q 3 r-+ ( ) E of one complex variable gives rise to an equation w I( ) = 0 in volving two complex variables. The function 1 in 10.1.6 satisfies e f( z ) 0 and the function in 10. 1 . 12 satisfies ( ) o. Each equation is an instance, for some subset £ of and a function F £ 3 (w, ) r-+ F( w , ) E of an equation of the form F w ) = 0 in which w is replaced by resp. Just as
-z
g
=
:
z
=
z C,
I(z) { (x, y) : ( x, y) E ffi? , x (1 y2 ) ! } � { (x, y) : (x , y) E ffi? , x2 + y 2 - 1 0 } , the inclusion { ( w, z ) : w - I (z) o } � { ( w, z ) : F ( w, z ) = o } ( ,z
_
=
=
=
when w is replaced by 1 can obtain. Both sides of the latter inclusion describe parts of a CAY, hence of a W, hence of an R. In the cited illustrations, if ( b) E £, F is representable at and
a, near ( a, b) by a power series L 00
'TTl,n=O
CTnn (w
a,
the series converges absolutely near ( b) , 00
rTL , n=O 00
clef ", Cr' (w ) ( z - b) n . =
� n=O
- a) Tn m(z - b) n . Since
408
00
Chapter 10. Riemann Surfaces
( a) -j. 0, there is a power series L dk (w - a) k such that near k=O a, F (w, � dk (w - a) k ) 0 (v. 5.3.43). If F(w, z) �f w - z2 and a b 1, then C (1) -2 and when w is near 1 there are two power series If C
1
=
=
=
1
=
(
Z = ±2 l + �2 (W - l) + m 21( � ) (W - l) ' +
) �f ±P(I, w)
F[w , ±P(I, w)] 21Tiot . The continuing (P, D(I, lr) along [0, 1] e results in (- P, D(I, lr). The same (P, D(I, In and (-P, D(I, lr). F(w, z) �f w 2 - z2 (cf. 10.1 .33), then C (1) 2 and z = ±[(w - 1) + 1] [�f ±P(I, w) ±w]. In this instance, both P and -P are entire and analytic con tinuation along "I : [0, 1] 3 t e 2 1T it of either of (P, D(I, lr) or (- P, D(I, lt) does not result in the other bnt ouly in the func tion element with which the continuation was begun. The CAY associated with (P, D(I, lr) differs from that associated with (-P, D(I, ln· When F(w, z) L cmn (w - a) m (z - bt , a series converging rn.n=O at and near (a, b), as a subset of ([2 , { (w, z) : F(w, z) O } may be regarded as the graph Qp of the equation F(w, z) = o. The topology of ([2 induces a topology on Qp . When (I, Q) is a func tion element such that for w in Q, F[w, f(w)] 0, for each w in Q, [w , f(w)] E Qp . The CAY associated with (I, Q) corresponds such that the curve "I : 3 t r-+ CAr is engendered by O n the other hand, if 1
=
=
=
r-+
00
=
=
=
to a ( possibly proper ) subset of Q F . Consequently a CAY, a W, and more generally an R may be viewed as a ( possibly proper ) subset of some Qp . Owing to the method of topologizing a CAY, a W, or an R, these subsets, like Qp itself, are for the most part locally conforrnally equivalent to 1U. However, as the consideration of the case reveals, Q F and one of the corre sponding CAY, W) or R are not necessarily homeomorphic. The global topological character of CAY and W is determined by the process of analytic continuation whereby any two neighborhoods may be connected. When analytic continuation is attempted at
F( w, z) �f w2 - z2
Section 10.3. Covering Spaces and Lifts
409
some neighborhood of OF , not every other neighborhood of OF is necessarily accessible. For example, the graph OF of is a connected = set, namely the union of two complex intersecting at Nevertheless their equations, = engender two (dif ferent!) complete analytic functions CAF± that lie in disjoint components of the sheaf S, v. 10.2.8.]
F(w, z) w2 - z2 stra'ight lines w ±z,
(0, 0).
1 0 . 3 . Covering Spaces and Lifts
p)
The triple (W, Q, consisting of a W-structure W and its associated region Q resp. local homeomorphism W r-+ Q is an example of a conforming to the general pattern (X, Y, p) described below. The contents of 10.2.18 - 10.2.25, treating the topological properties of a Riemann surface, are central to the discussion. To simplify the presentation, unless the contrary is stated, each topological space introduced below is assumed to be a curve-connected, locally connected ( whence locally curve-connected ) , and locally simply connected Hausdorff space.
p
triple
covering space
:
10.3.1 DEFINITION . A covering space tr'iple (X, Y, p) CONSISTS OF TOPO LOGICAL SPACES X AND Y AND A MAP p : X r-+ Y SUCH THAT: a) p IS A CONTINUOUS SURJECTION ; b) FOR EACH IN Y AND SOME OPEN NEIGHBORHOOD N(y), p- l [N ( y )] IS THE UNION OF PAIRWISE DISJOINT X-OPEN SUBSETS, EACH HOME OMORPHIC TO N(y) (N(y) IS evenly covered). FOR A CURVE [0 , 1] r-+ Y AND A POINT A IN p- l b(O)] , A lift of through A IS A CURVE ;:y : [0 , 1 ] r-+ X SUCH THAT ;:y(0) A AND p o ;:Y [ 10.3.2 Remark. Conventionally, X is a covering space of Y. The DEFINITION above is most convenient for the purposes below. Y
=
"I :
= "I .
"I
Alternative definitions, some more and some less restrictive, can be found in the extensive literature of topology.]
10.3.3 Example. a) As remarked above, a W-structure W is a covering space of the underlying region Q. b) For any topological space X , (X, X , id ) is a covering space triple. c) For X
(_ �f (0, 1) 2 , Y d_ �f ( 0, 1), and p : X 3
(
a, b) r-+ a E Y,
Chapter 10. Riemann Surfaces
410 P
-1
which is not homeomorphic to a union of pairwise dis = x joint X-open subsets homeomorphic to ( X, is a covering space triple. 10.3.4 Exercise. If X is a covering space of and is a covering space of Z, then X is a covering space of Z. 10.3.5 Exercise. If X is a covering space of and for some is homeomorphic to each component of
( a) a (0, 1),
Y, p) not Y Y Y y E Y, N(y), p - l [N(y)] N(y). 10.3.6 THEOREM. FOR A COVERING SPACE TRIPLE (X, Y, p) AND A CURVE "I : [0, 1] Y SUCH THAT "1 (0) �f a, IF A E p-l (a), THERE IS A UNIQUE LIFT ;:y : [0, 1] X "I THROUGH A. 10.3.7 Note. The result 10.3.6 is central to much of what [ f-t
f-t
a:
of
follows, in particular to 10.3.12.]
of ;:Y. N(A) p I N(A) p[N(A)] �f N(a) t l (0, 1] s tl , "Is(t) �f "I(st), N(a) N(A) N(a) "I(s) E N(a). p s t "Is A. ;:ys �f "Is s ;:Ys "Is A, f � 0, [0, t Il [0, 1, N b,,(l)] N p, N [;:y,, (1)]. ( 1], {a < s < (} hs(l) E N b,,(l)]} , and ¢" "Ie, is a lift of "Ie, through A. Thus a i- sup S, a contradiction: a = l. Uniqueness of ;:Y. If i] is a second lift of "I through A and
PROOF. Existence Some neighborhood is open and curve is a homeomorphism. Hence connected, and is open and curve-connected. Consequently, for some in if O :::; < then Thus, if ¢o is the f-t local inverse of and < then ¢o is a lift of through If S is the (nonempty) set of for which there is a lift of through S then a > and S = a] . C S, is connected, sup S If a < some curve-connected neighborhood is homeomor phic, say via ¢,,' an b,,(l)]-local inverse of to a curve-connected neigh borhood For some in (a, 1,
0
:
'*
0
E T i- 0, and continuity considerations imply T is closed. If t E T, N [i](t)] (= N [;:Y (t)]) is p-homeomorphic to a neighborhood N [1] (t)] (= N {p[i](t)]}) . Some neighborhood N(t) is mapped by "I into N[1J (t)]. If r E N(t), then p [i](r)] = p [;:Y (r)]. Since p I NF(t)] is a homeomorphism, it follows that r E T: T is open. Because [0, 1] is connected, T = [0, 1]. D
then ° some
Section 10.3. Covering Spaces and Lifts
411
In the following paragraphs there is described the construction of the
universal covering space X for a topological space Y. For a fixed point ao in Y and a (variable ) point a in Y, the set Sa of all curves "I : [0, 1] r-+ Y for which "1(0) ao and "1(1) a is decomposable =
=
into equivalence classes {Sa ,OI } OlEA for the following equivalence relation. Curves "I and 15 are equivalent ("I ev a 15) iff: "1(0) = 15(0) = (both curves start at "1(1) = 15(1) = (both curves end at ) for some continuous F [0, 1] 2 3 { x , t } r-+ Y,
ao a
•
•
•
ao); a;
:
F(x, 1) = F(x, O) = F(O, t) = "I(t), F(l, t) = r5(t), t E [0, 1] ,
ao,
a,
("I and 15 are homotopic ) .
clef
There emerges the set X = {Sa p } OlEA ' An element A of X is determined by: a point of Y; a curve "I such that "1(0) = and "1(1) = equivalence class of "I is A.)
a
•
•
ao
a
a. ( The homotopy
If A E X, there is in Y a ( unique) such that some homotopy equiv alence class of curves starting at and ending at engenders A. Hence there is defined a map p X 3 A r-+ E Y. The set X is given the topology T for which a neighborhood base is the totality of all neighborhoods as described next. For a point A Sa ,OI in X, a "l in A, and a neighborhood N( ) , a neighborhood N( A ) consists of the union, taken over all b in N( ) of the set of all homotopy equivalence classes of curves that are products "115 of "I and curves 15 such that 15(0) = 15(1) = b, and 15* c ) In brief, each point of Y gives rise to a set of homotopy equivalence classes. For each point b in some neighborhood N ( ) of there is a set Sb of homotopy equivalence classes, each consisting of all pairwise homotopic curves for which the curve-images start at pass through thereafter remain in N( ) and end at b. The neighborhood of A is
ao a
:
a
�f
a
a,
N(a .
a,
a
a a,
ao ,
a,
N(A) �f
U Sb . b EN( a )
Thus a neighborhood N(A ) of an A in X is determined by: •
a curve "I such that "1(0) =
=
ao and "1(1) a �f p(A);
a,
Chapter 10. Riemann
412 •
Surfaces
a neighborhood N(a) . The set X as described above is the for the space Y. 10.3.8 Example. If Y = 'lI' , then Y is not simply connected. The infi nite helix X � { (x, y, z ) : x = cos 27rt, Y = sin 27rt, z = t, - 00 < t < 00 } is the universal covering space of 'lI'. For (X, 'lI',
universal cov ering space
p
:
X 3 (cos 27rt, sin 27rt, t) r-+ cos 27rt + sin 27rt E 'lI',
i
p) is the covering space triple.
10.3.9 THEOREM. a) THE UNIVERSAL COVERING SPACE X IS A HAUS DORFF SPACE; b) (X, Y, IS A COVERING SPACE TRIPLE; c) X IS CURVE CONNECTED , LOCALLY CONNECTED (HENCE ALSO LOCALLY CURVE-CON NECTED ) , AND LOCALLY SIMPLY CONNECTED . PROOF. z, there a) If W and Z are two points in X and w exist disjoint neighborhoods N ( w) and N ( z ) . Hence the corresponding neighborhoods N(W) and N(Z) are disjoint. = there are curves 'Yw and 'YZ On the other hand, if w = connecting and w and such that:
p)
�f p(W) -I p(Z) �f
def p(W) p(Z),
ao
'YW : [0, 1] r-+ Y, 'Yz : [0, 1] r-+ Y, 'Yw (O) = 'Yz (O) = 'Yw (l) = 'Yz (l) = 11) , 'YW is not homotopic to 'YZ .
ao,
Moreover, w lies in an open, curve-connected, and simply connected neigh borhood N(w) . The pairs bw , N(w)] and b z , N(w)] determine neighbor hoods V(W) and V(Z) of W and Z. If E E V(W) n V(Z), there are curves I' : [0, 1] r-+ Y resp. 15 : [0, 1] r-+ N(w) for which =
ao, 1'( 1) w resp. 15(0) = w, 15(1) = p (E) and such that 'YE �f 1' 15 determines E. Since E E V(W) n V(Z), the ho 1'(0)
=
motopy equivalence classes bE } , bwr5 } , and bzr5 } are the same, i.e., br5 } = bwr5}
=
bzr5 } , bw }
=
bz } ,
a contradiction, since 'YW and 'Y Z are not homotopic. The remaining axioms for a topological space are directly verifiable, particularly in light of 1. 7.2. b) If A E X and (A) = any neighborhood N(A) is mapped by onto some neighborhood of is open.
p
a, a: p
p
Section 10.3. Covering Spaces and Lifts
413
If A E X, p(A) = a, and N(a) is a neighborhood in X, N(a) may be assumed to be simply connected. Then N (a) and some curve "( such that "((0) = ao and "((I) = a determines a neighborhood N(A) and, by definition, p[N(A)] = N(a) : p is continuous. If A E X and p(A) = a, because Y is locally simply connected, for some curve-connected open neighborhood N(a) , any loop containing a and contained in N(a) is null homotopic in Y. The neighborhood N(a) and some curve "( such that ,,((0) = ao and "((I) = a determines a neighborhood N(A). If W and Z in N(A) are determined by curves "(8 and "(( and if p(W) = p(Z) b [E N(a)] , then "(8 and "(( are homotopic, and thus W Z: p is injective. As an injective, continuous, and open map, p I N (A) is a homeomor phism. If A -I W E p- l (a) , N(W) is the associated N(a)-induced neighbor-
�f
=
�f
hood of W, and K E N(W) n N(A) , then p(K) C E N(a). If the curves that determine K, W, A are resp. "((, "(1] , ,,(e, then "(( and "(1] are homotopic, "(( and "( e are homotopic, whence "(1] and "( e are homotopic. Thus A = W, a contradiction. Thus different N (a )-induced neighborhoods are disjoint: p [N (a)] is the union of pairwise disjoint homeomorphic copies of N ( a). c) If 1]s : [0, 1] 3 t f-t Y is a set of curves, each starting at ao and de pending continuously on the pair (s, t), for each s, 1]s determines a point As in X: p ( As ) = 1]s ( l ) . If N (As ) is a neighborhood of As, there is some curve-connected open neighborhood N [1]8 (1)] to which is associated a neighborhood U ( As ) contained in N ( As ) . The map p I N (As) as IS a homeomorphism. If I hl is small, 1]s + h ( l) E N [1J s ( l ) ] , whence
-1
�f
depends continuously on s. In particular, if "( determines some A in X, i.e., if ,,((0) ao, "((I) = a, then "(s : [0, 1] 3 t f-t ,,((st) E Y depends continuously on the pair (s, t) and thus As depends continuously on s. The curve ;:Y : [0, 1] 3 s f-t As E X con nects A to the point Ao corresponding to the loop : [0, 1] f-t ao that starts and remains at ao : X is curve-connected. Since p is a local homeomorphism, the local properties of Y are local properties of X: X is locally connected and locally simply connected. D 10.3.10 Exercise. The universal covering space X, modulo homeomor phisms, does not depend on the choice of ao . [ 10.3.11 Note. For a given curve-connected space Y and its uni versal covering space X, there is an action, described next, of the group 7r l (Y) (v. 5.5.4) on the universal covering space X. Al though X does not depend on the choice of ao and 7r l (Y, Yo ) does As
=
f£
Chapter 10. Riemann
414
Surfaces
not depend on the choice of Yo , the discussion below is phrased in terms of some and a particular Yo.
ao
For ao fixed in Y and A in X, there is a curve "I starting at ao , ending at "1(1) and engendering the homotopy equivalence class that is the element A of X. If p E 7r dY, "1(1)], then p is represented by a loop 1] starting and ending at "1(1). The product 1]"1 is a curve starting at ao and ending at "1(1). The homotopy equivalence class A of 1]"1 is the result of the action of p on A: A = p . A. Since the constructs employed are independent of the choices ao and "1(1), P . A is well-defined in terms of p in 7rl (Y) and A in X.
�f a,
fixed in X , the set O(A) �f { p . A : p E 7r l (Y) } is the orbit of A. The projection p A r-+ a carries each point of the orbit of A into a. The set of orbits and Y are in bijective
For A 7rl (Y)
0
:
correspondence via p. The customary notation for this situation is Y X/7rl (Y).] rv
Paraphrased, the next result asserts that lifts of homotopic curves are homotopic and that their starting and ending points are the same. It is not only intuitively appealing but is the basis for a number of important conclusions, e.g., 10.3. 14- 10.3.17. 10.3.12 LEMMA. IF a) (X, Y, p) IS A COVERING SPACE TRIPLE; b) Ao E X, p (Ao )
t
(s, t)
=
ao
;
c) F : [0, 1] 2 3 r-+ "Is ( ) E Y IS CONTINUOUS ON [0, 1] 2 , i.e., GENER ATES A HOMOTOPY BETWEEN "10 AND "1 1 ; AND d) "Is(O) == "18 (1) THE LIFTS ;;;0 AND ;;;1 THROUGH Ao ARE HOMOTOPIC, AND END AT THE SAME POINT: ;;;0 (1) = ;;;1 (1) B.
ao ,
�f
==
b;
PROOF. Each "Is has a unique lift ;;;s through Ao. Naturally associated to F is F [0, 1] 2 3 r-+ ;;;s ( ) E X. Because X and Y are locally homeomor phic, the technique used in the PROOF of 10.3.6 applies: F is continuous and the curves ;;;0 and ;;;1 are homotopic. By definition, the connected sets ;;;o ([0, 1] ) resp. ;;;1 ([0, 1] ) are contained in the sets p- l ( o) resp. p- l (b) . Thus ;;;o ([0, 1] ) = Ao resp. ;;;1 ([0, 1]) is a point: ;;;0 (0) = ;;;1 (0) = Ao and ;;;0 (1) = ;;;1 (1) = B. D :
(s, t)
t
totally disconnected
a
10.3.13 THEOREM. EACH UNIVERSAL COVERING SPACE IS SIMPLY CON NECTED (cf. 10.2.37c) ) .
Section 10.3. Covering Spaces and Lifts
415
PROOF. For a point p in Y there is the homotopy equivalence class P (in X) of loops beginning and ending at p and loop homotopic to the constant curve l1 : E X is a t r-+ l1(t) == p. If ;::; : r-+ loop beginning and ending at P, for each ( ) E Y and = p(P) = p. For each in the curve ( ) = po
[0 , 1] 3
[0, 1] 3 7 ;::; ( 7) 7, p o ;::; (7) �f 1' 7 [0, 1], ;::;s : [0, 1] 3 7 r-+ ;::; ( S7) E X
;::; (0)
1' 0
s
is a lift through P-by virtue of 10.3.6, the unique lift through P-of the curve = p. If ;::; ( ) then r-+ E Y, and i.e., is the homotopy equivalence class of q = p o ;::; ( )p the curve in particular, P, which is the homotopy equivalence class of ;::; = l1. r5 p l1 is also the homotopy equivalence class of ( = D The next results show in what sense the universal covering space X of a space Y is universal.
I's : [0, 1] 3 7 I'(S7) (Qs), Qs I's(l) �f s I's, �f
I's(O)
s
�f Qs,
s
1'1 1'):
0
10.3.14 THEOREM. IF Y IS SIMPLY CONNECTED AND (Z, Y, ) IS A COV ERING SPACE TRIPLE, Z AND Y ARE HOMEOMORPHIC. PROOF. By definition, is continuous, open, and surjective. The next lines show that is also injective (whence a homeomorphism). If {p, q} c Z, there is a curve r : r-+ Z such that = p and r(l) = q . If (p ) = ( q) y, then r l' is a loop, and, since Y is simply connected, l' is null homotopic. The (unique) lift of l' through p is r, and the unique lift through p of the constant map r5 r-+ (p ) is l1 r-+ p. Thus 10.3.12 implies p = q. D a
a
a
a
a
�f
[0, 1] �f
r(O)
a 0
: [0 , 1]
: [0 , 1]
a
10.3.15 THEOREM. IF X IS THE UNIVERSAL COVERING SPACE OF Y AND (Z, Y, ) IS A COVERING SPACE TRIPLE SUCH THAT Z IS SIMPLY CON NECTED, Z AND X ARE HOMEOMORPHIC. PROOF. Each homotopy equivalence class in Z is mapped by into a ho motopy equivalence class in Y and thus to an element of X. Thus, W denoting the universal covering space of Z, there is a map F W r-+ X. If: a) is the basis of the construction of W; b) and is the basis of the construction of the universal covering space X, c) 10.3.12 implies, by abuse of language, that each homotopy class in Y lifts to a unique homotopy class (through of Z: there is a map G X r-+ W. Direct examination of the maps reveals that F o G and G F are the identity maps. By the same token, both F and G are continuous: W and X are homeomorphic. 10.3.14 implies that W and Z are homeomorphic. D 10.3.16 Exercise. The universal covering space X of a space Y may be characterized as the simply connected covering space of Y. a
Yo
Zo
o·
zo)
0
:
a
(zo) �f Yo; :
Chapter 10. Riemann Surfaces
416
10.3.17 Exercise. For a given space Y, the set C of all covering spaces is a poset with respect to the order: Xl >- X2 iff for some P12, (Xl , X2, P12) is a covering space triple. Relative to >- , C has a maximal element. This maximal element is the universal covering space of Y. The results 10.3.6, 10.3.12, and 10.3.15 apply when Y is an analytic structure AS (with its attendant set of function elements and region Q) and X is its universal covering space S. More generally, the same results apply when Y is a Riemann surface and X is its universal covering space. In the S-AS context, a denoting the projection of S onto AS, there are two covering space triples, (AS , Q , p) and (S, AS, a) .
If Y is a Riemann surface with its attendant atlas A and X is the universal covering space of Y, for X there is an atlas B such that with respect to B : a) X is a Riemann surface; b) the map p X r-+ Y is locally biholomorphic. = y , and p is a chart at y, a) If N(x) is a homeomorphism, ( ) , can serve as a chart ( ), 0 at b) If l ( ) , ;3 is a second such chart at then 10.3.18 Exercise.
:
[Hint: x.
p(x) {N(y), ¢} {N x ¢ p} �f {N x a} {N x } x, a ;3- 1 E H {a (N(x) n ;3 [Nl (x) J) } .J
I
0
Any two simply connected covering spaces of AS are holomorphically equivalent. An important consequence of the simple connectedness of S is a con siderable generalization of Riemann's Mapping Theorem. The latter may be stated as follows. If Q � e and Q is simply connected, Q is holomorphically equivalent to 1U. From this point forward, the context is a Riemann surface R with its attendant complete atlas � defined via canonical charts n i.e., charts such that ( n ) = 1U. When 0 < < 1 and is a chart, -;; l (D(O, a denoted . By extension, the complement of the confor < R\ mal disc < is 2': < and a punc tured conformal disc is Vn . In the spirit of the same : 0< 0 and if O (z) + In Izl E HaIR [D(O, rt] ' then 0 2': G. Hence the Green's function at is G. a
For C): Sections 5.8 and 5.9 and 6.5.8 provide the basis for the argument.
Section 10.6. Miscellaneous Exercises
427
�f
10.6.11 Exercise. If R is compact and K { z 1 z 1 :::; 1 } is a ( closed ) conformal disc on R, r < 1, f E Ha ( R \ { z : I z l 2': r } ) and I f(z) 1 :::; M, Stokes's Theorem and Green's formula are applicable:
r
fa ( K )
* df
=
r
fa (R\ K )
* df =
r
fR\ K
d * df = o.
(10.6.12)
The items C1)-C8) below are devoted to establishing (10.6.12) when R, which is not hyperbolic, is also not compact. C1) For the approximants Wn to Uf in the PROOF of 6.3.15, if {p, q} e N, 8p+ q Wn 8PH Uf then converge uniformly on compact subsets of Q to �x� y �x 8q y [Hint: 5.3.35 and 6.2.25 apply. ] C2) If R is not compact it contains relatively compact Dirichlet subregions Q such that K is a relatively compact in Q. For each such Q there is a function fn in Ha IR ( Q n { z : I z l 2': r }) and such that In ( z ) =
{ t(z)
if I z l = r if z E R \ Q '
The set of all such fn constitutes a Perron family F ( parametrized by the set of relatively compact regions Q containing K). If F = sup fn :F and I z l 2': r, then f(z) = F( z). [Hint: The Maximum Principle and the fact that R is neither compact and nor hyperbolic apply. ] C3) If gn E HaIR ( Q n { z : I z l 2': r }) and
{I
clef
if l z l = r gn (z) = 0 if z E 8 ( Q ) '
clef
the set 9 of all gn is, like F, a Perron family. If G = sup gn and 9 I z l 2': r, then G(z) = 1. C4) In F resp. 9 there are sequences {fn} nEN resp. {gn} nE N associated with relatively compact regions Qn and converging together with their derivatives uniformly on compact sets to F resp. G and their deriva tives, v. C2). C5) gn t:J. fn - fnt:J.gn = 0, n E N , ( v. 5.8. 18).
1
nn \ K
�
C6) If 0 < p < r < and Up E Ha IR [R \ D(a, p t ]' while up I Ca p = 1R(f) ( ) up is the solution of Dirichlet's problem for the boundary values 1R(f) then for k (r ) as in 6.5.7, max D r O up :::; k ( r ) .
R\ (a , )
Chapter 10. Riemann Surfaces
428
1 C7) If 5 > rn ..l- 0, for each n there I. S a sequence {Unrr,} m E N such that {Unm } ,n EN ::) {un+ l.m } m EN and for fixed n, Unm converges uniformly on A (0; rn , 1). For every positive r, the diagonal sequence {u mm } m E N converges uniformly on A(O; r, 1) to a function U harmonic in A(O; r, 1); the Maximum Principle implies convergence is uniform on A(O; r, (0 ) : {U mm } m E N converges uniformly on A( 0; 0, (0 ) to a function U harmonic in A(O; O, oo) . , C8) The argument in 6.5.7 leads to I [u � ( f )] (t �il! ) I :::; k(r) 1�t which converges to 0 as t -+ O.
-
( )
For D): w
10.6.13 Exercise. For fixed charts z resp. at a resp. b, if R is hyper bolic, Green's functions Ua resp. Ub exist. If R is elliptic or parabolic, there are functions Ua resp. Ub harmonic in R \ {a} resp. R \ {b} and such that are harmonic near a resp. b. resp. Ub(W) - � ua ( z) - � � (ua)x - i ( ua)y meets the require In either event the function f (Ub)x - i (Ub)y ments of D). [Hint: If Ua is a Green's function, for S me h holomorphic near 0, is harmonic u ( z) + In I z l = �[h(z)]. Similarly, if ua ( z) - �
(�)
(�)
O
(� )
(�) �(g) . h(z) - � resp.
near 0, for some g holomorphic near 0, ua (z) - �
=
In those respective cases, (ua)x (z) - i (ua)y (z) = 1 ( ua)x (z) - i (ua)y (z) = g(z) - 2 : the numerator resp. denomiz nator of f has a pole at a resp. b . ] For E) : 10.6.14 Exercise. a ) For a in R, there is in H (R) an f that in each chart z, u ( z) = G(z) + In I z l , v. 10.4.8. If
�f U + i v such
F(z) � e xp [G(z) + In I z l + iv(z)] ,
I I
clef
z = - In I ha(z) l · then In I F(z) 1 = G(z) + In I z l and G(z) = - In F(z) b ) I ha l < 1.
Section 10.6. Miscellaneous Exercises
429
ha(b) - ha(z) IS . holomorphlc . on R and I ¢ < 1 c) For b fixed, ¢ : R 3 Z r-+ I 1 - ha ( b » ha(z (v. Section 7.2 and Chapter 8) . d) If Ord f (a) �f n, off Z(¢) , u �f - '! ln l¢1 is harmonic and positive off n the zeros of ¢. e) If ( is a chart at b, then :F consisting of functions w such that: el) w is subharmonic and nonnegative off b; e2) supp (w) is compact; e3) w (() + In 1(1 is subharmonic at b, is a Perron family. f) u ( ( ) � G(() �f - In I hb l and hb (a) � 1¢(a) l * � 1¢(a) l . g) 1- == l. hb h) Z(¢) = b and ¢ is injective. Riemann's Mapping Theorem implies the Uniformization Theorem for hyperbolic Riemann surfaces. For F): 10.6.15 Exercise. 1 Fl) For some meromorphic f, g Z r-+ f(z) - - is holomorphic near a, f :
Z
is holomorphic off each neighborhood of a, 'iR(f) �f u is bounded, and u(a) - 'iR o. F:r some meromorphic 1, g Z r-+ 1(z) - is holo
(�)
�
:
=
morphic near a, and f is holomorphic and bounded off each neighbor hood of a. Near a, both f and 1 are injective. If b is near but different 1 1 cl f from a, then g clef = are h 1omorp h'lC ff b and -g =e f - f(b) f - f(b) while b is a simple pole of g and ?j. Some linear combination ag + fig is holomorphic and bounded. Since R is not hyperbolic, ag + fig is con stant on R: for some Mobius transformation T, f = T Finally, 0
0
(1),
(1).
1 = if, 2s(f ) = -'iR whence f is admissible. F2) The end of the argument in Fl) implies that the set S of points b such that for some Mobius transformation T, f = T (fb) is both open and closed: S = R. F3) If f is admissible at a and f(a) = f(b), for a g admissible at a and con structed as in Fl), for some Mobius T, g = T(f), whence g(a) = g(b) . Since a is the unique pole of g, a = b. F4) If R is elliptic, i.e., compact, for an f admissible at a, f(R) is both compact and open, whence f(R) = Coo . b
Chapter 10. Riemann Surfaces
430
If R is parabolic, and f is admissible, f (R) c Coo and f is not sur jective. For some Mobius T, T o f(R) c C. If T o f(R) �C, then T o f(R) is conformally equivalent to 1U: R is hyperbolic, a contradic tion. 10.6.16 Exercise. The fundamental group 7r l (R) of a Riemann surface is finite or countable. [Hint: The result 10.2.10 and the construction of the universal covering space apply.] 10.6.17 Exercise. a) For some f in H (1U) , if Z E 1U, then 2 exp { [ J ( z W }
J ;
=1+
z,
1 Z and is holomorphic in 1U. b) Ana i.e., f is some branch of ln lytic continuations of f along the curves 'Y± [0, 1] 3 t r-+ ±1 + e2rrit lead to different function elements. c) The functionals 'Y± are homologous in Q � C \ ({l} u {-I}) . d) The Q� homotopy equivalence classes h+ } h- } and h- } h+ } are different, i.e., h+ } h- } h+}- l h_ r l -j. 1. Hence 7r l (Q) is not abelian. 10.6.18 Exercise. Without reference to the Uniformization Theorem, for any Riemann surface, the conjunction 5j 1\ IE is impossible. (Thus the listing of possibilities in the second and third columns of Table 10.3.1 is exhausti ve.) 10.6.19 Exercise. If Q c C, f E H (Q), and for each a in C and some nonempty open neighborhood N(a) , f may be continued analytically to a function element (fa , N( a ) ) , there is an entire function F such that F l n = f. 10.6.20 Exercise. When Q � [0, 1] 2 is regarded as a subset of C, if f E H(Q) and f[8(Q)] C lR, there is an entire function F such that F I Q = f. [Hint: The Schwarz Reflection Principle applies.] :
11
Convexity and Complex Analysis
1 1 . 1 . Thorin's Theorem
In a number of disparate contexts, e.g. , topological vector spaces, probabil ity theory (measure theory confined to measure spaces (X, 5, p,) for which X E S and p,(X) = 1 [Kol]) , von Neumann's theory of almost periodic func tions on groups as well as his theory of games [NeuM] , linear and convex programming, etc. , the role played by convexity is central. The following discussion attempts to illustrate that role in complex analysis. 11.1.1 DEFINITION. WHEN {Vd 1 < k < n IS A SET OF VECTOR SPACES, Fk E [0, (0) Vk , 1 :::; k :::; n, AND G E ilR � ¢ IN lRV, x " , x Vn IS (G; F1 , " ' , Fn )n CONVEX IFF WHENEVER 0 :::; tk , tk = 1 , AND G o ¢ IS DEFINED, k=l
L
G O ¢ (V l , . . . , Vn ) :::;
n
L tk Fdvk) '
k=l
[ 11.1.2 Note. In the context of Jensen's inequality (3.2.35), a function ¢ convex in the ordinary sense is (id ; id , id )-convex. When V1 Fl
=
�f U (X
clef In II II
p,
, p"
) V2 �f U' (X , p, ) , F2 cle=f In II II G cle=f In,
Vi E V;, i = 1, 2, f (v l , v2 ) �f
p' ,
I x Vl (X)V2 (X) dp,(x) I '
Holder's inequality states that ¢ �f ln f is (G; F1 , F2 )-convex:
431
Chapter 11. Convexity and complex analysis
432 2 7T n i n
� e-nSince 2 7T kt
1-e = ----,c;2-=-w'- = 0, for h in MVP [D( a, r t] n in N, � 1 - e� k =O and s in [O , r ) , � denoting approximately equal to, h ( a)
=
=
h
[n-l� :;;: ( 1
'
a + se
2 7r k i -n
rIr h (a + st) dT (t)
I1l'
= 0
1�
)
1
� :;;: h
k=O
2Wk,
(a + se-n- ) . (11. 1.3)
Thus, when = and � are read :::; , (11.1.3) may be viewed as a convexity property of h. By abuse of language, subharmonic func tions, may be viewed as a class of convex functions.] When X is a set, B, D c X, and I E jRx , the Maximum Principle in D relative to B obtains for I iff sup I (x) :::; sup I(x) . xED
xEB
11. 1.4 Example. The Maximum Modulus Theorem asserts, i.a., that if
D(a, r) C Q and I E H(Q) , the Maximum Principle in D � D( a, r t relative to IJ �f 8[D( a, r )] obtains for I II . 11. 1.5 Exercise. If D e B, the Maximum Principle in D relative to B obtains for all I in jRx . 11. 1.6 Exercise. If a function I in jRlR is convex or monotone, for every finite interval [a, b] , the Maximum Principle in [a, b] relative to the set {a, b} (consisting of the two points a and b) obtains for I . 11. 1.7 Example. If if ° :::; x :::; 27r I (x) �f { �Os x otherwise for every finite interval [a, b] , the Maximum Principle in [a, b] relative to { a, b} obtains for I in jRIR . However, I is neither convex nor monotone: the converse of 11.1.6 is false. 11.1.8 LEMMA. FOR A VECTOR SPACE V, A ¢ IN jRV IS CONVEX IFF FOR EACH A IN jR, EACH PAIR {x , y } IN V, AND EACH t IN [0, 1] , THE MAXIMUM PRINCIPLE IN THE INTERVAL [x , y] �f { z : z = tx + (1 - t)y, t E [0, 1] } RELATIVE TO {x , y } OBTAINS FOR ¢ [tx + (1 - t ) y] - At .
433
Section 11.1. Thorin's Theorem
The map ¢ in lRv is convex iff for each A in lR
sup {¢[tx + (1 - t )y] - A t } ::;; max{¢(y), ¢(x) - A }.
Oz carries QC onto A( a, r R) . Hence I 0 ¢ conforms to the hypotheses of 11. 1. 11. D [ 11.2.4 Remark. It is the method of proof of Thorin's Theorem rather than the theorem itself that leads to the last two results. In [Thl the author attributes his attack on his general theorem to Hadamard's original approach! 0',
:
:
For In M(x; I) resp. In M(r; I), one may substitute g o M(x; I) resp. g o M(r; f) when g is a monotonely increasing function con tinuous on the right. A notion of the strength of Hadamard's Theorem and of the last remark can be derived from the following considerations. The Maximum Modulus Theorem implies merely M(x; I) :::; max{M(a; I), M(b; I)}
Chapter 1 1 . Convexity and complex analysis
436
resp.
M(t; I) � max{M(r; I), M(R; I)} .
These inequalities do not imply the convexity of either M(x; I) or M I); a fortiori, they do not imply any of the convexity properties of In M(x; I), In M(t; I), g o M(x; I), or g o M( ; I).
(r;
r
The replacement of, e.g., M by In M, leads to a function like ¢ in 11.1.1.] 11.2.5 THEOREM. IF I E H(1U) ,
r E (0, 1 ) ,
p
E (0, 00 ) , AND
r;
a) ON (0, 1 ) , Ip(r) IS A MONOTONELY INCREASING FUNCTION OF b) ln lp(r) IS A CONVEX FUNCTION OF ln r. PROOF. a) If ° � r 1 < < the Maximum Modulus Theorem implies
r2 r,
that for some
r2 ei02 , � t I I (r 1 e 2""w i ) I P � � t I I (r2 e 2""W ' ei02 ) I P . k= l
k=l
b) Thorin's method of proof as exhibited in the PROOF of 11.2.2 and as used in the PROOF of 11.2.3 applies. D 11.2.6 THEOREM. (M. Riesz) IF
FOR THE ( BILINEAR ) MAP B : em
x
0',
f3
>
0,
Tn
n f en 3 (x, y) r-+ L L ajkXjYk � B(x, y),
j=l k=l
THE CONDITIONS 0, (J'k > 0, 1 :::; j � m , 1 :::; k � n, def def P = ( P1 , · · · , Pm ) , 0" = ( (J'l , . . · , (J'n ) ,
Pj
>
AND S THE SET OF ALL (x, y, P, 0" ) SUCH THAT m
L j IXj � :::; 1, j=l P l
and
n
L (J'k IYklfr :::; 1, k=l
Section 1 1 .2. Applications of Thorin's Theorem
437
THE LOGARiTHM OF Mu(3 �f
sup I B(x, y) 1 IS A CONVEX FUNCTION (x.y . p.CT)ES OF THE ( a , /3) IN THE ( OPEN ) QUADRANT Q �f { ( a , /3) a , /3 > o } . [ 11.2.7 Remark. Thus, if :
{ (I' , J), (1], () ) C Q , 0 � t � 1 ,
and ( a , /3) = t (1', J) + ( 1 - t)
(1], ( ) ,
(MO:(3 is a multiplicatively convex function in Q).] PROOF. The notations below are useful in the discussion that follows:
For >'1 , ), 2 real and fixed and (0'0, /30) in Q, owing to 11.1.11 and 11.1.12, it suffices to prove that the logarithm of
is a convex function of t on (0, 00 ) . If the real variable t is replaced in the right member above by the complex variable t + i u , each ¢j resp. 'l/J k is translated by U), l In rj resp. U), 2 In S k , and the value of the right member is unchanged:
If all the variabl�save t and u are fixed, Hadamard's Three-Lines Theorem is applicable to M (0'0 + )'It, /30 + ), 2 t) . Thus In M (0'0 + )'It, (30 + ), 2 t) is a convex function of t on (0, 00). D [ 11.2.8 Note. a) When 0' = 0 or /3 = 0, 1
1
a
0
clef 00 or
1
-
/3
=
1
-
0
cle=f 00.
Chapter 1 1 . Convexity and complex analysis
438
n Tn Concordantly the conditions L I Xj I i- :::; 1 or L (J"k I Yk I � :::; 1 are k= l
j =l
interpreted I Xj I :::; 1, 1 :::; j :::; m , or I Yk I :::; 1, 1 :::; k :::; n. The ar gument given for 11.2.6 remains valid when a = 0 or f3 = 0, v. 3.2.6. b ) M. Riesz showed that for the bilinear map BIR : ]R2 x ]R2 3 (x, y) f-t (Xl + X 2) Y l + ( Xl - X 2 ) Y2 , 1 when a = 2 ' the logarithm of the minimum of
is concave. Thorin extended M. Riesz's result by showing that if the logarithm of the maximum is concave [Th] . Thus, o < a :::; if B is restricted to ]RTn ]Rn the conclusion in 11.2.6 is invalid un less 0' + f3 z 1 and {a, f3} C [0, 1] ' i.e., unless (a, f3) E Q, v. Fig ure 11.2.1.]
�,
X
11.2.9 Exercise. In terms of j= l
an equivalent formulation of 11.2.6 is the following.
(0,
I,
1)
o
1)
L-------�-- a
Figure 1 1 .2.1.
Section 11.2. Applications of Thorin's Theorem
The map M(3 : Q r-+
(x,y,p,CT) E S V (
0, AND
U (G, p,) ,
*
THEN Il f g il T :::; Il f ll p . Il g ll q · PROOF. If {j, g} C Coo (G, q the inequalities *
Il f g ll oo :::; Il f ll p' . Il g ll p , Il f g ll p :::; Il f l l l . Il g ll p , *
imply that T,q is a map both from LP to Coo (G, q (c L 00) and from LP to LP. The multiplicative convexity of No:(3 and 1 1 .2.12 apply. 0 The M. Riesz Convexity Theorem has many consequences. In particu lar, it leads to the theorems of Hausdorff/Young, and F. Riesz in functional analysis. Since their proofs use the M. Riesz Convexity Theorem for which the proof above involves complex function theory, they appear at this junc ture. The setting is a locally compact abelian group G equipped with Haar measure p,. The central facts relevant to the current discussion are Pon trjagin's Duality Theorem (v. 2) in Section 4.9) and its ramifications in functional analysis. These are treated in detail in [Loo, N ai, We2 ] .
441
Section 1 1.2. Applications of Thorin's Theorem
a) According as G is discrete, compact, or neither, i.e., locally compact and neither discrete nor compact, !he dual group of G is compact, . discrete, or neither. Furthermore, = G. b) To each Haar measure p, for G there corresponds a dual Haar measure for O. bI) If G is discrete, L 1 (G, p,) C L 2 (G, p,) . b2) If G is compact, L 2 (G, p,) C L 1 (G, p,) . b3) If G is discrete, say G = {g,X L E A ' and e ,X cle=f X{ } ' E A , the Fourier transforms tG., ). E A, are a CON for L2 b4) If G is compact, {¢,X L E A is a CON in L 2 (G, p,) , and f E L 2 (G, p,) , then 2 ) (1, ¢,X ) 1 2 = l f(x W dp,(x) (Parseval's equation) . ConG 'x E A 2 versely, if L i c,X 1 < 00 , for some f in L 2 (G, p,) , (I, ¢,X ) = c,X . 'x E A The last is a generalization of the classical Fisher-Riesz Theorem (v. 3.7.14) . c) If G is neither discrete nor compact, neither of the inclusions
0
11
0
g"
\ /\
(0,11) .
l
need obtain. However, L 1 (G, p,) n L 2 ( G, p,) �f S is a dense subspace of L 2 (G, p,) . If f E S, then 1 E L 2 and
(0, 11)
(Plancherel's Theorem) . Thus there is definable an extension, again denoted -, to L 2 ( G , p,) of the Fourier transform and
is an isometric isomorphism (v. 4.9.6). d) The statement in a) is logically equivalent to the conjunction of the statements in b) and c): {a)} {} {{b)} 1\ {c) } } . e) If 1 :::; p < 00, by abuse of notation, P � Coo(G, q Coo (G, q (the subspace generated in Coo( G, q by functions arising from convolution of functions in Coo(G, C) ) is II lip-dense in LP(G, p,) . Furthermore, if and the Fourier inversion formula f E P, then 1 E L 1 *
(0,11)
442
Chapter 11. Conve�ity and complex analysis
is valid. 11.2.14 Example. The discussion below is based on items a) -e) above and the interpretations and extensions in 5) of Section 4.9. For the map
the previous observations imply
,
1 8 (I, g) 1 � Il f ll l . Il gll l = I l f 112 , 1 8 ( I , g) 1 � II f l12 . II gl1 2 .
11 � 1 2
The multiplicative convexity of No:(3 implies whence if 1 � p � 2, then
11 � l p' � Il f ll p·
11 .2.15 THEOREM. (Hausdorff/Young and F. Riesz) a) IF 1 < p � 2 AND THE CON{¢n} nE ]\/ (DEFINED ON [0, 1]) CONSISTS OF UNIFORMLY BOUNDED FUNCTIONS (FOR SOME M, II ¢n lloo � M, n E N, ) FOR f IN
THERE OBTAIN
{ cn �f 1 1 f(x)¢n(X) dX } nE]\/ E £P' , AND (� len IP' )
b) IF {cn } nE ]\/ E £P , FOR SOME f I N U' ([0, 1] , q ,
1 1 f(x)¢n(X) dx {e2mrit}
=
Cn, n E N, AND Il f ll p'
�
1
[7
�
(� ) len I P
Il f ll p . 1
p
PROOF. The discussion of 1 1 .2.13 suffices for the CON consisting of the functions n EZ appropriately re-enumerated as a sequence {¢n } nE]\/ ' For the general CON the observations 00
2 L I cn l � Il f ll� n =l
len I
�
(Bessel's inequality),
M ll f ll l ' (M n-free, )
443
Section 1 1.2. Applications of Thorin's Theorem
LP LLP L
n 2 3 f r-+ { cn } n E N ' regarded as a map from imply that the map T the function space n 2 to the function space eN , is one to which the multiplicative convexity of No: f3 applies. 0 [ 1 1 .2.16 Note. As the counterexamples below demonstrate, the condition 1 1, c E lR \ {O}, and f( x ) �f
OCJ
L n� (In n)f3 e 2mrix ,
n= 2
icn In n
'i
then f is continuous but if p > 2, then b) The trigonometric series
f
I
�icn ln n
n= 2 n 'i (In n)f3
I
P' =
00 .
(11.2.18) is not a Fourier series, i.e., there is no f in L 1 such that if m -j. 2 n otherwise [Zy] . Nevertheless, if q
>
2,
00
L1 ( n1� )q
n=
1, the study of functions holomorphic in a region of en IS simplified by the introduction of specialized vocabulary and notation. 445
446
Chapter 12. Several Complex Variables
When Q l , . . . , Qn are regions in c., their Cartesian product is a polyregion . When each Qk is an open disc D (a k , rk t , Q is a polydisc . ( + ) n �f clef �f When a = ( a 1 , • • • , an ) E Z , z - ( Z l , . . . , Zn ) , a - (a l , . . . , an), and r clef = ( r l , . . . , rn ) , Z
= Zl
a clef
"'1
a! = a l ! ' "
clef
'"
. . . Zn n , an ! ,
aa
=
clef
a"" , . . . , "' n ' clef I a I = 0' 1 + . . . + an ,
ra clef II ri'" n
==
i= l
t ,
a"' n clef al a i a clef a"" . . "' n = a J = "' 1 . "' 1 aZ l ' aZn aZ l ' " aZna n '
12.1.3 Exercise. For a polydisc b.(a, rt, how do aD [b.(a, rtl
and a [b.(a, rtl
differ? Many of the theorems about functions in CC have, when n > 1, their natural counterparts for functions in c.c n • There follows a systematic listing of these counterparts. ' 12.1.4 THEOREM. IF J E C[b.(a, r) , C] AND, AS A FUNCTION OF zk , WHILE THE OTHER COMPONENTS OF ( Z l , . . . , Zn ) ARE HELD FIXED, THEN IN b.(a, rt,
1
1 J(z) = -( 27r i ) n ao[� ( a,r ) Ol ( Z l
J(w) dW · · · dwn . ) · · · ( Zn - w n ) l wI -
12.1.6 COROLLARY. IF J E H (Q) , THEN J E C = (Q, q .
(12.1.5)
447
Section 12. 1. Survey
12.1. 7 THEOREM. IF I E H (Q) , K (Cn ) THERE ARE CONSTANTS Co. SUCH THAT
3
K c Q, AND 0 ( Cn )
3
U ::) K,
12.1.8 COROLLARY. IF {In} nE]\/ C H (Q) AND FOR EACH COMPACT SUBSET K OF Q, In I K � 1 1 K ' TH EN I E H (Q) . 12.1.9 COROLLARY. THE n-VARIABLE VERSION OF VITALI'S THEOREM (5.3.60) IS VALID. 12.1 .10 THEOREM. IF I E H [b.(a, rt] ' THEN
THE SERIES CONVERGES UNIFORMLY TO I IN EVERY COMPACT SUBSET OF b.(a, r t . 12.1.11 THEOREM. ( Cauchy's estimates ) IF I E H (b.(0 , rt) AND I I(z) 1 � M IN b.(a, rt, 12.1.12 THEOREM. ( Schwarz's lemma ) IF I IS HOLOMORPHIC IN A NEIGH � ORHOOD OF b.(O, r) AND II(z) 1 � M IN b.(O, r) , FOR SOME k IN Z+ AND k ALL z IN b.(O , r) , I I (z) 1 � M .
I�l
12.1.13 THEOREM. ( Jensen's inequality) IF b.(0, r) C Q, I E H (Q), AND In II(z) 1 dA 2n � In 1 1(0) 1 · 1(0) -1 0, THEN A 2n b. 0, r ) ) �(O,r) [ 12.1.14 Remark. When n > 1, Jensen's inequality implies that if I E H(Q) and I =t=- 0, then
(\
1
A 2n ({ z : z E Q, I(z)
= O } ) = O.
Results like ( ) above are valid for holomorphic maps between complex analytic manifolds, v. [Gel].] *
When n > 1, there are theorems that have no nontrivial counterparts when n 1. =
448
Chapter 12. Several Complex Variables
12.1.15 THEOREM. (Hartogs) IF n > 1, Q c en , I E en , AND AS A FUNC TION OF EACH VARIABLE Zk , AS THE OTHER COMPONENTS OF ( Z 1 , Z ) ARE HELD FIXED, I IS HOLOMORPHlC, THEN I E H (Q) . [ 12.1.16 Remark. The contrast between 12.1.15 and 12.1.4 deserves attention.] .
. ·
,
n
n
PROOF. (Sketch) a) The formula in (12.1.5) implies that if II ri > ° and I I I
i= 1
n
is bounded in a polydisc Q �f II D (0, ri t , then I is continuous, whence 12.1.4 applies.
n
b) An argument relying on Baire categories shows that if II ri > ° and
i= 1 Q, then D(a, r ) contains a polydisc l1 such that l10 -j. (/) and I I I is bounded in l1 . c) Mathematical induction, Cauchy's estimates (12.1 .11) (for the case of n 1 variables) , and 6.3.39 conclude the proof [Ho] . 0 D(a, r)
c
-
n
12.1.17 THEOREM. (Polynomial Runge) IF Q �f II Qk IS A SIMPLY CON k= l
NECTED POLYREGION IN en AND K IS A COMPACT SUBSET OF Q, EACH I IN H (K) IS UNIFORMLY APPROXIMABLE ON K BY POLYNOMIALS. [ 12.1.18 Note. If b is sufficiently small and positive and Q is the polyregion [D(O, 1 + bt] 2 D(O, bt 3 ( Z 1 ' Z2 , Z3 ) , the map X
is injective and holomorphic. Nevertheless, on F(Q) Polynomial Runge fails to hold [Wer] .] On the other hand, there are no generally valid counterparts to remov able singularities (cf. 5.4.3) nor to the phenomenon of natural boundaries as exemplified by 7.1.28 and 7.1.29. 12.1.19 THEOREM. IF n > 1, Q IS A REGION CONTAINING THE BOUNDARY OF D(a, rt, AND I E H(Q) , THERE IS IN H (D(a, rt) A UNIQUE 1 SUCH THAT ll nnD(a.r)O = I l nnD(a.r)o ' PROOF. (Sketch) For z in D ( a, r t the formula 1
I ( z ) = 27ri
-
�
1
I w - a n l =r
I ( Z 1 , . . . , Zn - 1 , W ) W - Zn
dw
449
Section 12. 1. Survey
o
defines the f as described.
12.1.20 COROLLARY. IF n > 1 , r > 0, AND a IS AN ISOLATED SINGULAR ITY OF AN f HOLOMORPHIC IN D(a, rt \ {a} , THEN a IS A REMOVABLE SINGULARITY OF f. ALL ISOLATED SINGULARITIES ARE REMOVABLE, CF. 5.4.3 . 12.1 .21 THEOREM. IF 1 � k
0,
'. Z l = . . . = Zk = 1, Zk +1 < 1 + E , . . " Zn < 1 + E } ,
II
I I I I S, ::)
I I
D(O, 1t u r 1 = . . · = rk = 1, rk+ 1 = . . · = rn = 1 + E, r cle=f ( r 1 , . . · , rn ,
Q
)
AND f E H (Q), THERE IS IN H [Q U D(O , rn A UNIQUE f SUCH THAT ll = f · n PROOF. ( Sketch ) The formula
defines a function as described. o There arises the question of characterizing an open set in en as a domain of holomorphy, i.e., roughly described, an open set U for which some f in H (U) has no holomorphic extension to a proper superregion U1 , cf. 7.1 .29. 12.1.22 DEFINITION. AN OPEN SET U IN en IS A domain of holomorphy IFF FOR no OPEN SETS U1 , U2 : a ) (/) -j. U1 C (U2 n U) ; b) U2 I S CONNECTED AND U2 ct U; c ) WHENEVER f E H (U) THERE IS IN H (U2 ) A ( NECESSARILY UNIQUE ) h SUCH TH AT f l u! = h l u! ' 12.1 .23 DEFINITION. FOR A COMPACT SUBSET K OF AN OPEN SET U, THE H (U) -hull OF K IS
Ku �f
{z : z
E Q, U E H (U)}
'*
{ I f(z)1
� s�p
I f (z ) l } } .
[ 12.1.24 Note. The set Ku is closed but need not be compact.]
450
Chapter 12. Several Complex Variables
12.1.25 THEOREM. IF U IS OPEN IN en , THE FOLLOWING ARE EQUIVA LENT: a ) U IS A DOMAIN OF HOLOMORPHY; b ) IF K IS A COMPACT SUBSET OF U, THEN Ku IS RELATIVELY COMPACT IN U; c ) SOME f IN H (U) HAS NO ANALYTIC CONTINUATION BEYOND U, i.e. , THERE ARE NO U1 , U2 CONFORMING TO a) AND b ) IN 12.1.22. 12.1.26 COROLLARY. IF U IS CONVEX, U IS A DOMAIN OF HOLOMORPHY. 12.1.27 DEFINITION. THE SET OF ALL POLYNOMIALS IS Pol. A Runge domain U IS A DOMAIN OF HOLOMORPHY SUCH THAT IF f E H (U) , THEN f IS UNIFORMLY APPROXIMABLE ON EACH COMPACT SUBSET OF U BY ELEMENTS OF Pol. WHEN K IS COMPACT,
WHEN K = K, K IS polynomially convex. 12.1.28 THEOREM. A DOMAIN OF HOLOMORPHY U IS A RUNGE DOMAIN IFF FOR EACH COMPACT SUBSET K OF U, K = Ku . Further details can be found in [Ho] and [GuR] . They provide exten sive treatments of the results cited above and relate them to the theory of partial differential equations, the study of Banach algebras, complex ana lytic manifolds, etc.
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Symbol List The notation a.b. d indicates Chapter a, Section b, and page d. a.e.: 2.2. 60
A \ B: for two subsets of a set X, { x A(a; r, R) : 5.4. 234 A(a : r, R; 8, 4;) : 5.4. 234 [B, F]c: 3.3. 105 Bn ( f ) : 3.2. 100 BP (x, r): 1.2. 10 BW: 3.4. ' 112 (Bft : 3.4. 112
x E A, x tJ. B }, 1.1. 3
c:
1.1. 5 10.2. 395 Cabed: 8.4. 359 c : 2.2. 69 C: 1.1. 4 cn : 1.1. 5 C= : 5.6. 252 C* : 4.9.192 C: 8.4. 359 CF: 3.5. 129 C k (C, lR): the set of functions f having k continuous derivatives, 5.1. 203 C = (C, C): the set of functions f having k continuous derivatives, k E N, 5.1. 203 Co: 1.2. 13 Co(X, C) : 3. 1. 90 Cu (r): 8.2. 344 Coo(X, C)) : 2.3. 61 Coo (X, lR) : 2.3. 76 Cubed: 8.4. 359 Ck , Tn : 4.6. 165 CON: 3.2. 97 Cont (f) : 3.7. 134 Conv (A): of a set A, the intersection of the set of all convex sets containing A, 3.7. 136 cos: 2.4. 80 div : 6.3. 287 DLS: 2.1. 50 D: 2.2. 59 D( a, r)o: 5.1. 203 Sz :
455
456
D: D (L 1 ) , 2.2. 56 D(L): 2.2. 56 D( f ) : 9.1. 369 Rt : 5.5. 234; 10: 5.4. 251
D(O,
Symbol List
det : determinant, 4.7. 183 dP,' dP: 5.8. 261 diam (S): 1.2. 15 dim : dimension, 3.4. 112 Discont (a ) : 10.3. 416 [V] C : 3.3. 104 X- I : 3.5. 117 (Xo , . . . , Xn ) : 1.4. 21 [Xo , . . . , Xn ] : 1.4. 21 (X, d) : 1.2. 8 (X, Y, p) : 10.3. 409 yX : the set of all maps f : X H Y, 1.2. 7 { z : Izl < r }: 10.3. 416 3: 1.2. 9, 10.2. 396 ,,: 10.2. 396 ,6(X): 3.7. 136 * "( : 2.4. 79 "( "'A J: 1.4. 20 "( '" J: 1.4. 20 "( "' F, A J: 1.4. 20 ;y, r: 5.5. 237 l1u: 5.3. 226 Ja : 2.1. 55 J(a, b) : 5.6. 252 J(F) : 1.2. 14 (: 5.9. 269 8 : 5.6. 251 I-l([a, b] ): 4.10. 196 XK: I-l 4.4. 154 1-l0K : 4.4. 154 * I-l , I-l* : 2.2. 68 3.2. 97 n + : 5.7. 254 7r(x) : 5.9. 269 7rl (Y, Yo) , 7rl (Y): 5.5. 245 p(T) : 4.7. 178 P(x): 3.5. 122 L: 2 : the Riemann sphere, 5.6.251 a (B, B') , a (B', B) : 3.4. 112 a�k) : 1.4. 27 v:
459
Symbol List
460
T: 3.5. 127 Q: the equivalence class containing the well-ordered set of all ordinal num bers corresponding to countable sets, 1.1. 5; a region contained in C, 5.1. 204 w ( k ) , *w ( k ) : 5.8. 258-259 N o : 1.1. 5 (0, Q) : 2.2. 67 #: 1.1. 5 aj; (X) : 4.7. 182 aXj l � i .j 0 and )"�(E) < 00 , for some finite set 11 , . . . , !rn of pairwise disjoint elements of I, ).. � E \ < 15.
---
---
(
4.6. 169
--- Theorem: 5.3. 228
U ;=Jj)
w
WEAK, WEAK': 3.4. 112 weaker, weakest: 1.2. 6 wedge product: 5.8. 261 WEIERSTRASS, K . : 3.5. 126, 5.4. 230, 7.1. 310, 7.1. 312, 8.6. 366 WeierstraB Approximation Theorem, 3.5. 126 elliptic function: 8.6. 366 product representation: 7.1. 312 WeierstraB-Casorati Theorem: 5.4. 230 well-behaved: 4.8. 186 -ordered: of an ordered set X, that each subset has a unique min imal element, 1.1. 5 Well-ordering Axiom: Every set X may be well-ordered, i.e. , there is an order --< such that for any two elements x and y of X, x --< y or y --< x and for any subset Z of X, there is in Z a (unique) z such that for any other element z' of Z, z --< z' (every subset has a least element). 1.5.
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32
489
Glossary/Index WEYL, H . : 10.2. 395 winding number: 5.2. 208 W-structure: 10.2. 391 x
x-FREE: independent (free) of x, 1.6. 37 X-holomorphic, X-meromorphic: 10.2. 404 x-neighborhood: 1.7. 39 x-section: 4.4. 153 y
YOUNG, G . C . : 4.6. 177 YOUNG, W . H . : 4. � . 190,
11.2. 386, 11.2. 442 z
ZAREMBA, S . : 8.1. 342 ZERMELO, E . : 4.5. 160 Zermelo-F'raenkel system of axioms: [Me] , 4.5 160 zero: 5.4. 231 of order or multiplicity n o : 5.4. 232 Z ORN, M . : 1.2. 13, 1.5. 32 Zorn's Lemma: In a poset (r, --< ) , if every ordered subset has an upper bound in r, for each "I in r, there is in r a maximal element f.-l such that "I --< f.-l, 1.5. 32