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O
and
N,
a positive integer such that p(f) ;; C 1\ fl\ V
for every
f
in
H(U;F).
N
Hence
sup p(f) ;; C.N fE J
and this completes the proof.
If
Corollary 2.44
U is an open subset of a LocaLLy convex space
is a locaLLy convex space and every bounded subset of locaLly bounded then
T
0'
T
T~
and
Ul
(H(U;F)
E, F
is
,TO)
have the same bounded subsets in
v
H(U;F).
In particular Notation
is the bornoLogicaL topology associated with
TO
etc. in place of
Corollary 2.45
E and
Let
bounded subsets of
(H(U)
of E then
=
Proof
H(U;F)
linear space.
f(K)
,TO)
H(U;F ) a
~y(U;[),
H(U;[),
= =
•
F
etc.
be LocaLly convex spaces.
Let
where
=
Fa
(F,a(F,F')).
B be the unit ball of
(epof) epEB ep(f(K))
lies in
H(U).
If
Thus
J-
F
Fe
F
is a normed
f E H(U;F ). The a K is a compact subset of U then
is a bounded subset of
is a weakly bounded subset of
strongly bounded.
If the
are LocalLy bounded for every open subset U
We may assume without loss of generality that
epcf(K)
o
If the range space is the field of complex numbers we write
H(U), HHY(U)
set j.
T
[
and let
for every
ep
in
F'.
Hence
and by Mackey's theorem it is
is a bounded subset of
(H(U),T
O
)
and
50
by
our hypothesis, it is locally bounded. Hence for each E, E U there exists an open set containing E, such that supllepofll v 0 be arbitrary. We first k l L(ol) C U and next choose 81 and 8 2 , 81 > 1
8 82L(01) 1
is also a compact subset of
U.
p(f)
for all
f
in
inequalities.
H(U;F) Hence
where
c(8 l )
is derived by using the Cauchy
Hence
80
Chapter 2
is a
p(f)
continuous semi-norm on
H(U;F)
we can find a positive integer sup \'L 00 _ f£1 n-N+l
Let
J
and since N
is
T
o
TO -bounded
and
13 >1
2
such that
- - II II ~nf(O) n! L('\)
~ E/ 4 .
be the symmetric n-linear form associated with dnf(O)/n!. n!
If
f E H(U;F)
and 0
>
then
0
for any non-negative integer Since
sup fd
II
dnf(O) n!
n.
II
O
n
and
m.
m. If
V is any neighbourhood ~
c(V) II fllv for every of K such that
11K V open
relation which
f
.-.J
where
and
g
We denote by
f,..... g
if there exists a neighbourhood
W of
K on
are both defined and H(K;F)
the resulting vector space of equivalence classes
and the elements of H(K;F) are called holomorphic germs on K. If f is an F - valued holomorphic function defined on an open subset of E which contains K then we also denote by f the equivalence class in H(K;F) determined by
inductive limit
f.
The natural topology on lim (H(V;F)"w)
H(K;F)
is given by the
(the inductive limit being taken in the
--+
v::n
Vopen category of locally convex spaces).
for
If F is a normed linear space we let Hoo(V;F) = {f £ H(V;F) ;11 fllv < oo} V open in E and on this space we define a topology by means
of the norm II II V' HOO(V;F) is a normed linear space which is complete if F is a Banach space. Using the same equivalence relationship we easily see that
85
Holomorphic mappings between locally convex spaces
u
H(K;F) .
V'::'K
V open
If
Lemma 2.53
K is a compact subset of a locally convex space
E and
is a normed linear space then
F
lim
lim
H(K;F)
--+
If
Proof
V~K
V:l K
V open
V open
E which contains
V is an open subset of
natural injection from
Hoo(V;F)
the identity mapping from
lim
-
--+
V:JK V open
V:::>K V open is also continuous. lim
Conversely, if
(Hoo(V;F), II Ilv)
K then the
into (H(V;F),T ) is continuous and hence W oo (H (V;F), II II V) into lim (H(V;F) ,TW)
then for each
p
is a continuous semi-norm on
V open,
V'::>K
there exists
c(V»
0
---+ V~K
V open such that f
E
p (f) ~ c (V) II f II V
H(V;F)' Hoo(V;F)
then
Hence the restriction of
to
lim
(H(V;F),T ) w
---+
f
in
Hoo(V;F).
If
and the same inequality holds.
Ilfllv p
ported by the compact subset norm on
for every H(V;F)
K of
V.
is a
T
Thus
P
w
continuous semi-norm is also a continuous semi-
and this shows that the two topologies coincide
V::) K
V open on
H(K;F)
and completes the proof.
It follows that
H(K;F)
is abornological space if
F
is a normed
linear space and an ultrabornological (and hence a barrelled) space if is a Banach space. then
H(K;F)
If
E
is a metrizable space and
F
F
is a Banach space
will be a countable inductive limit of Banach spaces and
hence a bornological
DF-space.
Thus we see that the space of germs will
always have some good topological properties since it is an inductive limit and indeed the main topological problems connected with
H(K;F)
are those
generally associated with inductive limits (as opposed to those connected with projective limits) such as completion, description of the continuous semi-norms (sometimes we only need a description of sufficiently many
86
Chapter 2
continuous semi-norms) and a characterization of the bounded sets. encounter all of these problems in later chapters. selves to characterizing bounded sets when Let
E
= lim Ea
We shall
Here we confine our-
E is metrizable.
be an inductive limit of locaUy conVex spaces.
The
----+
a
inductive limit is said to be regular if each bounded subset of contained and bounded in some
B
E. a
K be a compact subset of a locally convex space W be a convex balanced open subset of E. Then
Lemma 2.54 let
E is
{f£H(K+W); IlfllK+w
Proof
E and
Let
Let
{fa}aEA
is a cLosed subset of H(K).
(I}
be a convergent net in
is a Cauchy net in
H(K)
which lies in
show that
{fa}a£A
subset of
K+W
(H(K+W),T O) '
L C K+pW.
By using the Cauchy inequalities we see that
then there exists a real number
~~~ I am:~)
If
p,
B.
L is a compact
O - - 1 }. 1 n n+ n + Z Z
Example 2.57 V = {z E (;; z n Let
fn E H(V n
U
Let
E
Wn )
be identically one on
Vn
and identically zero on Wn
91
Holomorphic mappings between locally convex spaces If
K=
I {n} n:;:l
for all
n.
u {O}
then
K is a compact subset of
{fn}~=l
The sequence
and
([
fn
E:
H(K)
satisfies condition (a) of proposition
2.56 but is not bounded by proposition 2.55. We shall use proposition 2.56 and its method of proof again in a later chapter.
The following result follows easily from the corresponding result
for holomorphic functions on open sets.
If
Proposition 2.58 E and
K is a compact subset of a locally convex space
is a complete locally convex space, then
F
closed complemented subspace of H(K;F)
for every positive integer
In discussing holomorphic functions on a compact subset locally convex space complete.
E we may always suppose that the space
"E
If not, let
is a
(@ (mE;F),T) w
denote the completion of
E.
If
m.
K of a E
is
V is a convex
balanced open subset of E then the natural restriction from Hoo(K+V) oo into H (K+V () E) is a bij ecti ve isometry and hence the spaces H(K ) and E H(Kg) are isomorphic as locally convex spaces, (KE(respectively Kg) is the set
K considered as a subset of
E
(respectively
E)).
We shall also need hypoanalytic germs in later chapters and the definitions are exactly as one would expect.
If
K is a compact subset of a locally convex space
locally convex space then on ence relationship on which both
f
~
and
by g
U
E and
F is a
HHy(V;F) l,)e define the equivalKCV, V open f'" g if there exis ts a neighbourhood W 0 f K
are defined and coincide.
We let
U ~y(V;Z'F) Kev
.rvI
V open
and the elements of this vector space are called F-valued hypoanalytic germs on
K.
On
~y(K;F)
we define the locally convex inductive limit topology
-
lim
V:;>K , V open
We have used an inductive limit of topology on
H(K;F).
TW
topologies to define the natural
It is not known if the
as a projective limit of the spaces
H(K;F).
TW
topology can be recovered
This problem is rather impor-
tant and appears to be intimately related to the localization problem
Chapter 2
92
mentioned previously:
is
Tw
a local topology?
inquiry, we can define a "new" topology on a locally convex space and Tn·
It is conjectured that
If
Definition 2.59
and
F
H(U;F),
Tn
and
TW
always coincide.
U is an open subset of a locally convex space
F is a locally convex space then the
H(K;F)
One easily sees that TO:i Tn:i TW on give examples of situations in which T n
2.60
Show that
topology on
Tn
E
is
H(U;F)
K ranges over the
where
u.
compact subsets of
EXERCISES
U an open subset of
a locally convex space, which we denote by
defined as the projective limit of
§2.5
Following this line of
In chapter six, we shall coincide.
T
W
for any open subset
H(U;F)
HG(U;F)
dimensional space
H(U;F). and
E and any locally convex space
F
U of an infinite
if and only if
E ,:::: «(N) . 2.61*
(E,t f )
Show that
is a topological vector space if and only if
E
has a countable algebraic basis. 2.62 E onto
If
E and
F
F are vector spaces and
for any 2.64*
If
x Let
E. Let f for every 2.65*
is a linear mapping from
is an open mapping.
--+
2.63
TI
show that
E and
F are vector spaces and
in
Yl' ... 'Yn
E,
E E - n(K) .
H(U;F)
if and only if
f
is a
normal mapping. 2.74* in
Let
(~n)n
E be a locally convex space and let
be a sequence
E'. (a)
If
is a Frechet space or a ~:fm. space show that
E
I:=l( ~nf
H(E)
E
for every
x
~n (x)
i f and only i f
in
-+ 0
as
n -+ 00
E. oo
(b)
If
E = (E,a(E,E'))
only if of 2.75*
If the locally convex space
is
such that
if
EB
E
E,
dnf(x)
F E
(the vector span of
B in
a?>
is a Banach space and CPM(nE;F)
for some
x
f
E
in
TM
HG(U;F)
If
f
TM
complete locally convex spaces, and
HG (U; F)
continuous show that
f
where
E ~(U;F).
U is a
E
is
1M
complete.
complete locally convex show that
f
E ~(U;F)
U and every positive integer
2.76*
E
B normed by the gauge
then we say that
F
being
if and
contains a fundamental system of
U is a connected T M open subset of the
space
H(E)
is finite dimensional.
(~n)n
bounded sets
E
is a null sequence and the vector span
(~n)n
of B) is complete for every If
I n=l(~ n )n
show that
TM
open subset of f
E x F, E
n. and
is separately Mackey
95
Holomorphic mappings between locally convex spaces
2.77
If
U is an open subset of a locally convex space
locally convex space, show that for each
f
in
HG(U;F)
E and
F
is a
the following are
equivalent: (a)
f E H(U;F)
(b)
for each
(respectively in
E;
U
,II
dmf(E,) )"" m=O m! is an equicontinuous (respectively locally bounded) family of mappings. 2.78
E be a Banach space and let
Let
sequences in
(H(E) "0) be defined by L is well defined and holomorphic.
Show that If
f
E
be the space of all null
endowed with the sup norm topology.
E'
L : Co (E ')
Let
2.79
Co (E')
--->-
H( IT E)
where each
is a locally convex space, show
of
A and
a
Al
such that
-.J
f
= fOIT
If
"" n Ln=l (- '"
If Te E E' for each a in some directed set Q and Bn is the projection of B in En then liTanllBn
= S~pITa(X~)
Hence, since En T~
,
IITaIlB' for all n. I -< l2 n cr n is a complemented subspace of E,
in (E n )13' as a ' for each n ifTa , o i n E '. 13 This shows that {(E )13'}n is a Schauder decomposition for E '. n 13 To complete the proof it suffices to notice that 0
00
H%morphic functions on balanced sets
119
1
i12 for every T in E'. Corollary 3.14
An/i-absolute decomposition of a locally
convex space is a shrinking decomposition. Further properties of associated topologies and Schauder decompositions are given without proof as the need arises. For further details we refer to the Notes and Remarks at the end of this chapter. Apart from the immediate applications of the above results which we give in this chapter we will also find that the techniques developed to obtain these results are useful in studying holomorphic functions on certain nuclear sequence spaces. §3.2
EQUI-SCHAUDER DECOMPOSITIONS OF (H(U;F) ,T). In this section U will denote a balanced open subset of
a locally convex space E and F will denote a Banach space (Most of the results, however, can be extended to arbitrary quasi-complete locally convex spaces). Proposi~ion
If U is a balanced open subset of a locally
3.15
convex space E, F is a Banach space, f EH(U;F) and a
=
(an)n EJ then g
l:'"
a
n=o
H(U;F)
E
n
Let K c:. U be compact.
We may suppose without loss
of generality that K is balanced, then there exist A>l and V, a convex balanced neighbourhood of zero, such that A(K+V) C U and
Ilf II A( K+ V)
M
< '" •
By the Cau c h y in e qua l i tie s
Ildnf~O)
lI:qK+V) .:::. M for all n. n. Since (an) E ~ there exists C > n
0
such that
lanl .:::. C f;A)n for all n. Hence II gil K+V
.:::.
;="'0 ani 1
o such that
for every f P
~ denote the algebraic closure of V in E, i.e. 'V
{x E E; AX E V for
V
2. A
0
0
are such that
IT(f)1
~ CTIK(f)
HN(E)
then the proof of propositionl.47 O for all n. Since K is a neighbourhood oo ~oo (BT ) < cE 1 n = 2c < 00 this ~=o TI~KO n n=o 2 and the image by B of an equicontinuous is a bounded subset of Hoo(V) Also,
for every f
in
shows that
TIKo(BT n )
of zero in implies BT
(E',T ) O E H(O)
~
and
subset of (HN(E),TIo)'
for some neighbourhood V of 0 in
since the Borel transform is an isomorphism
on each space of n-homogeneous nuclear polynomials, we have shown that B is a well defined injective linear mapping. Conversely let A ;: {g
E
o Hoo(K );
~:o
An lid
;f O) ~KO
~
I} where K is
c
Chapter 3
138
a compact subset of E.
For each g in A there exists, by
proposition 1.47, a unique sequence of linear functionals,
n
CT ) 00 T n n=o' n nand then
e: WNC E),'11 ) ' ,
o
ITnCP) I 2. c'llK(P Z;OO
n=o
IT
'n
(d f(o))1 n n!
such that BT
~NCnE).
for every P in
Since Ii=l
n
~
a
144
Chapter 3
K eKo + {hN K
0
CI.
nYn; fl>Nlanl a
co +{ff=l
CK 0
+q~:l
2a(N+n)Z
ISn l
::.
ISn l
::.
co
C U0 +{fl= I Sn x n'. co
I
5 fl=l
+ I V + I Ii +
CK
4
4
Ia n I
::.
1}
ISn l -< I An I for all n}
S x n n
I C K0 + 4 V +
1}
co 2 Za(N+n) Y + ; L I N n
n
"" CK 0 +{ff=l Sn x n
-
-
n
Let E and F be locally convex spaces and let U be a
3.93
balanced open subset of E. TE:{T
Show that
(H (U; F) ,T) ,
,
T , T r } , is semi-Montel if and only if (H(U;F) ,T) is o w u T.S. T complete and C&CnE;F),T) is semi-Montel for each non-negative integer n.
3.94
Let E be a locally convex space.
Show that
(H(U) ,TO)
is complete for every open subset U of E if and only if (~(nE),T ) is complete for each non-negative integer nand o is T.S. TO complete for each convex balanced open
H(V)
subset V of E.
151
Holomorphic functions on balanced sets 3.95* that E
n
Let {E}
be a sequence of Banach spaces.
n n
(H(~"'l E ) n=
n
,T
0
Show
is a semi-Montel space if and only if each
)
is finite dimensional.
3.96*
If E is a locally convex space and f£HNCE)
dnf(x) £iPNCnE)
show that
for every x in E and every positive integer n.
Show, by counterexample, that the above condition on f£H(E)
is not sufficient to insure that it lies in HN(E).
Show also that HN(E)
is a translation invariant subalgebra
of H(E). 3.97*
Let E be a locally convex
linear space.
spac~
A function f £ HCE;F)
and F a normed
is said to be of
exponential type if there exist a continuous semi-norm a on E and positive numbers c, for every x in E.
Let
C such that
Ilf(x)
II
~ C exp
(ca(x))
Exp(E;F) denote the set of all
holomorphic functions of exponential type from E into F. Show that f = ~'" n=o
dnf(o) n1
£
Exp(E;F)
if and only if there
exists a continuous semi-norm a on E such that lim sup [ sup
{II
A
dnf(o) (x)
II;
1/
a(x) ::.. I}] In < "'.
n --+ '"
3.98
If E is a Banach space and f£ H(E)
f £ Exp(E;C)
=
Exp(E)
show that
if and only if the restriction of f to
each one dimensional subspace of E is a function of exponential type. 3.99
If E is a locally convex space show that the
mapping
An f = ~'" d ff o ) n=o n.
£ Exp(E) ----+l
!;'"
n=o
is a linear bijection. Using the above, or otherwise, describe a locally convex topology on Exp(E)
so that the above bijection is a linear
Chapter 3
152
topological isomorphism. Let E be a Banach space and let f and g be hOlomorphic
3.100
functions of exponential type on E. function on E show that h
E
If h
Let E be a locally convex space.
3.101
= fig
is an entire
Exp(E). An element f of
HN(E) is said to be of nuclear exponential type if there exists a convex balanced open subset V of E such that An
lim sup TIv(d f(o)) n -+
lfn
n E E' for aU n el>n E H(E) i f and only i f el>n (x) - - - + 0 as n---+ 00 n
n=o for every x in E Ci . e . el>n ---+
0
in the w* topology) .
oo
L
n eI> E H(E) then ,'" L. (eI> (x)) n z n converges n=l n n=l n for every x in E and z in [. By the Cauchy-Hadamard formula
Proof
If
in one variable nlim _ sup
~
Conversely if el>n(x)
,00
n
then f = L. el>n n=l
E
lei>
---+ 0
HGCE).
n (x)nll/n = lim
n-~
as n ---+
ro
I eI>
n (x)
I
=
O.
for every x in E
Since the nth derivative of f at 0
Chapter 4
166
is ~n and this is continuous we may apply theorem 2.28 to n
complete the proof. Example 4.6 Let E be a separable Hilbert space with oo Let f(L z e) roo zn for orthonormal basis (e )00_ , n n- 1 n=l n n n=l n all
L z e sE. n=l n n oo
\,00
Hence f = L n=l
at the nth coordinate of E.
n
~n
where
Since
~n
~n----+
is evaluation 0
as n ----+
in (E',cr(E',E)) lemma 4.5 implies that fSH(E). · 1 an d 1 ~m n ->-
sUPl1 00
dAnf(o)
__ II l/n
00
However
1·
n!
Example 4.6 shows that in infinite dimensions we have to distinguish between the "radius of pointwise convergence" and the "radius of uniform convergence".
A further concept
is the radius of boundedness which enters in a natural way and plays an important role in later developments.
Let U
be an open subset of a locally convex space E and let B be a balanced closed subset of E.
We let I f E is
a normed linear space and B is the unit ball of E then dB(~'U)
E.
~
is the usual distance of
to the complement of U in
Now let F be a Banach space and let fsH(U;F).
B radius of boundedness of f at
~,
sup {IAI;As,,~+ABCU, "f"~+AB
rf(~,B),
0
n = inf{dBCCU), (lim f(Olil/n) -l} n _ sup lIa n! B 00
If then
167
Ho[omorphic functions on Banach spaces
We first note that if E = U then dBCs,U) = + 00 and the above may reduce to ~ = = ~. This however says
Proof
00
that f is bounded and the Taylor series converges uniformly on s + AS for every A £( i f and only i f lim sup IlcinfCs) Il lln =0: n---+ oo n! B If
0
lal
- '"
0
as m
-+
00.
for all n sufficiently large, say n
2,
Hence for any sup IfCs+x)1 x£ ~B
0
0
xn }: = 1 ) when Re(x ) ;:. o . l
if x
=
(A,
0
.)
•
and y
(fl,o
• ),A,fl>o
.
This cannot happen on uniformly convex Banach spaces. Definition 4.15.
A Banach space E is said to be uniformly
conVex i f for every e: x,y e: E,
iixll = iiyii = I,
>0
there exists a 0
iix+yii ~ 2-0 We have
>
0
such that for
iix-yll ~ e:
172
Chapter 4
lp is a uniformly convex space for l- 0 as n I fex ) I > n for all nand
This contradicts the fact that
n
completes the proof. §4.2
BOUNDING SUBSETS OF A BANACH SPACE In the previous section we considered sets and regions
where a single function was bounded.
We now look at sets on
which every holomorphic function is bounded.
00
Holomorphic functions on Banach spaces
Let U be an open subset of a locally convex
Definition 4.17 space E.
sup
173
A subset A of U is said to be bounding for U i f = IIfllA < for every fEH(U).
I f(x) I
00
XEA
We shall use the term bounding set when the domain space U is easily understandable from the context.
Bounding
sets arise naturally in problems of analytic continuation, construction of the envelope of holomorphy and in problems concerning topologies on H(U). We begin by
collecti~
some simple properties of bounding sets.
Let U be an open subset of a locally COnVex
Lemma 4.18
space E and let F be a locally convex
~pace.
Then
(a)
every compact subset of U is bounding,
(b)
the closure of a bounding set is bounding,
(c) i f f £H(U;F) and A is a bounding subset of U then f(A) is a bounding subset of P (d) i f A is a bounding subset of U then IIfllA
Proof Let V
(a), (b) and (c) are obvious. {f n
£
with
II: then
')(~)A(xl)j(X2)n-j
j=o
+
£
By lemma 4.29
sup
we have shown Similarly we find
liP lis
lip IIs3
U
i =1
liP
U
>
2
1U
S4 ~
S
>
2-
12£
II
2 £.
I!pll ~ IlPll)n
since
Ip(x +A xl)1
I A 1::..1
By induction
This is impossible for all n
S. 1 <X>
Now let u
and hence we have completed the proof. (0,
n
.
.0,
1,
0
•
",""nth positive integer n and let A
=U
.)
for each
position
{un}'
A is a c los e d
n=l
bounded non-compact subset of l
Theorem 4.31
Proof
A
is a bounding subset of
Suppose A is not bounding.
exists an entire function
loo.
By corollary 4.19 there
f on l", such that
An Hence since as n
~
oo
Id
for every x in loo, we can choose
f~o) (x) I
n.
-+
0
(if necessary
Holomorphic functions on Banach spaces
181
by restricting f to Z",(S) ~ Z",} an increasing sequence of positive integers, I
(nj);=l' such that
dnjfn.( 0) (u. ) IV n j ! J J
The function g
>
-
i
2
>
0
for all j.
r"j=l
belongs to HCZ",) and
"n·
d Jg(o)(u.) nj
An.
1 for all j.
J E
1-s (nj
J
For each integer j let . d Jg(o) ~n
Z "') where An. J
Cet kl
n. ! J
1.
Choose
Sl infinite such that kl ~ Sl and sUJ;l
L
This is possible
IAI.s..1 o 0 there exists a finite subset J£ of N such
that for any finite subset J of N which contains J£ we have Ilx -
L
j £J
lp'
xje j II ~ £).
I < P < "',
and Co all have unconditional bases
and the finite product of spaces with an unconditional basis also has an unconditional basis.
The space of all
convergent series is an example of a Banach space (with a which has not an unconditional basis. The following result is well known and consequently we do not include a proof.
basi~
Chapter 4
184 Lemma 4.34
If E is a Banach space with an unconditional
(en)~=l,then the bilinear mapping from loo x
basis,
((~n)~=l'
given by
I''' n=l
xnen)
,
r'
~
n=l
E -- E
x e n n n
is well defined and continuous. The above property in fact characterises Banach spaces with an unconditional basis.
Lemma 4.34 allows us to renorm E with an equivalent but more useful norm. Lemma 4.35
(E,II II)
Let
(en)~=l) then the norm
unconditional basis,
III I'"
be a Banach space with an
III
sup
n=l
JC N
finite nd
I An I:::..
n xn e n
A
II
1
is equivalent to the original norm On E. assume that the given norm on E
Henceforth we shall satisfies
,,'"
~=l
III'"
x e n =1 n n
II
III
= sup J C N, J fin i ten E J
I A J' I:::.. xnen
E
E and
x
A
n n
e n
II
for all
1
in this case the bilinear mapping of
lemma 4.34 has norm 1. We now introduce some notation for the Banach space E with unconditional basis If 0:::"
m
S
i
, 2
:s.- i :s.-
m-I
m-1
and we let
B
.,S m- I
51 '
• ,13
13 1 '
(Cln)~=l
x B ={I
m
A.13.z.;tn Z.e: B,Aiel[
ill
m
1
1
i=l
1
5.
IAi
:s.-
+
52 13 1T S (B) 2 1
S
III Lemma 4.36
1f
1 (B)
z. e: E5 1 _ 1
l
1,
i
1
where 5 0 =0 and 5 m
such that
0
lire f) II 2. C'
. sup ~=l,
for every f in H(U)
Ilf Ilv .. ,r
(* *) .
Ki s
, i) d Let a i ="4 d(Ki,/\OU) for i = l, ... ,r an suppose
K. e E l ~
for i = 1, ... r if U f E. Then (*) and (**) imply that there exists C > l 2. Cl sup
lire pn ) II
i
C
=1, ..
such that
r
n sup (a ) lip n 111 K.+B i i = 1, ... , r ~
l
a
C
0
l sup i = 1,
... , r
(a. ) n ~
i
lip n III a.
sl Ki+Bl,l
(4 . 1 )
~
for all P
n
£
iF (n E) -and for all n.
Now suppose we are given m+l positive numbers Cm,Sl'"
"Sm'
a strictly increasing sequence of positive integers sl'"
.,sm_1 and y
lin Pn)
>
1 such that
II 2. C sup mi=l, ... ,r
s , K.+B 1 S I' a. ~
.,sm_l
h n III
. ,S
~
for all P
n
Sm+l
>
£
(lJ(n E) and all n.
and yl
0
>
m
(4 .2)
Then we claim that given
Y there exist Cm+ l
>
0
and sm
>
sm_l
such that . , sm
(4.3)
"Sm+l for all
P n
£
If>
(nE) and for all n.
Suppose otherwise. exists
P '
n
Then for every positive integer n there
a homogeneous polynomial of degree k , such that n
190
Chapter 4
sup i= 1, .. , r
IIT(Pn)ll>n
We first show sup k n
Otherwise, by taking a subsequence n
if necessary, we may suppose k
M for all n.
=
n
By lemma 4.36(b) 1
- )
1 K. 1 ('ti
-1
~i
-
B
and hence the sequence
sup i=l, ... , r
K +B
i
sl'· .. ,s _l's _l+n m m ~l' ... '~m+l
is a locally bounded subset of (fJ(ME)
n=l
and a TO bounded
subset of H(U).
II
Since
Pn
- - - - - - - - - , r - - - - - - - - - - - - - - ) II
T
1
sup i=l, ... , r
(y C!i)
kn
.
s1
IIPnlll K +B ~ ' C!. 1 1, 1
for all n this is impossible and hence sHP k
> n
... ,sm_l +n ~
... , m+l n
=
By
00.
taking a subsequence if necessary we may assume that
(kn)n
is strictly increasing sequence of positive integers. k s l+n P T\ Aj mand hence, by (4.4), there exists n j=o
L
Now
for
~ach
s
IT(A.
m l -
In
integer n, +n
)11
n
k n -+
00
o.
By taking a subsequence
n
kn-jn if necessary we may suppose ---k--n
---+
e: as n
~
00
Holomorphic functions on Banach spaces
For each n let L Let x =
L'"
x.e. 1 1
i=l
k i s ( nE) with L
£
n
E and 0 >
£
n
be arbitrary.
0
III'"
positive integer N such that
= P
n
191
Choose a
xieill < o.
Now
i=N
I
l+n A. mS
In
(x)
k
I (.n)L (L
I
s
.
+n
In x.e.) x.e.) 1 1. 1 1 1>sm_1+ n
(L
m-1 n i=l
I n
k Y1 k k -j n ( . n) I L ( ( Il x 11+ 1) n o n n In
TJXll+T
s +n \' m-1 (where Y1 = L x.e. i=l 1 1
)
jn
Y2 L
kn-jn
'6{llx 11+1)
)
k
.
n-Jn
I
I. 1 1 x. e. )
and Y2
1>sm_1+ n
k
( IIxll + 1) n
+
Now suppose sm_1+ n > N.
Y1 IllfxTIiT
+
Y2 0( Ilx 11+ 1) II
:s..
Then
0 11%1'1 + o (11x 11+1)
1.
Hence, by lemma 4.36 Y1
lTXTFT
Y2 + o(I~II+1)
£
(I m+ 1 i=l
1 )
t\
B
,sm_1,sm_1+ n
s l' . 13 , . 1
, 13 m+1
and, by lemma 4.37(a), 1 im sup n
:s..
-->- '"
lim sup
n-'"
sup 1= , .. ,r
. 1
1 k (a.y ) n 1
lip
(11x 11+1) y
1
sup i=l, .. ,r
ai
n
II
sl' "1'
K.+B Q 1
. ,sm_1, sm_1 +n . , 13 m+1
I
Chapter 4
192
n:i=lm
(1Ixll+l) OE 1
Y
sup
a·1
i
Since
>
However if f n=o is a '0 (and hence a '8) bounded subset of H(E) and An
II T(d ~fO)) II
consequently lim n-+
I; n
O.
Thus the above
00
kn-jn
gives a contradiction and so lim n-+
00
-k-n
~
0
as n
----+
00
We now consider this case. Since r is finite and fixed and the sequence (kn);=l is infinite we may suppose, from
(4.2)
if necessary, that there exists s
liT (A I. m - 1
+n
n for all n.
k
1 ) II .::. em ( y
ai )
n
and taking a subsequence
i,l~i~r,
such that
•
193
Holomorphic functions on Banach spaces 5
Hence lim sup n - ""
II
T (k
m 1 a. n IIA. I
1
(~)
.
1
h
n-Jn )
n
(by lemma 4.37(b)).
Sm
Hence lim sup n---->- '"
>
II
194
>
Chapter 4
lim sup n-
n
co
(k +1) n
>
Y
1
since k
>
n
Y
l/~
1
n and lim
o.
n-+
1
This is impossible since Y
>
and thus we have proved the
y
required step in our induction argument.
Aside
A simplified version of the above goes as follows;
if the induction step did not work then we could find k -j jn n n .l =.l .l , where "'n is evaluation at the nth n "'1 "'n coordinate, such that the sequence (fn)n did not satisfy
f
k -j If ~ ~
(4.4) . nth
k
E
>
0
then the rapid decrease of the
n
coordinate overcomes the geometric growth
first
coordinate so that In fn
kn-jnj k
of the
Otherwise
H(E).
so that the effect of the nth coordinate is jn negligible and In fn behaves like In ~1 In both cases n
-
E
0
we saw that this led to a contradiction.
We now complete the proof of the theorem. Let
(Yn)n denote a sequence of real numbers'Y n
such that ITn=l Y = y n (y-l)
sup xEK
Now using
Ilx II
1,
E such that
for i=l, ... ,r
i
(4.1) as the first step in the induction and
since (4. 2)
~
(4.3) we can find a strictly increasing
sequence of positive integers,
(s )'" l' n n=
and (C )'" 1 a n n=
Holomorphic functions on Banach spaces
sequence of positive numbers such that liT (P
n
)
II
for all P
£
n
(p(n E ) and all n.
Let K = sxB where S= (S)
n n
and
1 if n < s2 1 if i si ~ n
l such that AL is again a compact subset of U.
If W is any open subset of U
which contains AL then there exists a neighbourhood V of K such that ALCAVCW. Hence, for any f proposition 3.16
r" n=o
dnf(o) nl
£ H(U) we have, by
195
Chapter 4
196
II T (f) II
-
2. n
(aj)~=o lies in Co and, for each f in H(U),
The sequence we have
p (f) .::. Lt'" P j=o
j
(d f(O)) j!
.::.
I
n
-1
sill.
1
)..j
j=o
Aj
+
I'"
li
n=l
sup j=l, .. ,n Hence any
TW
d ~ (0) J !
I'"
Il
j=o
l
dj f
I K +-1 B
(0) j!
n
I
K+a.B. J
continuous semi-norm on H(U)
is dominated by
a semi-norm of the required type and this completes the proof. We now look at holomorphic functions on a countable direct sum of Banach spaces.
Properties of bounding sets
enable us to settle the "completion problem" for such spaces and the techniques developed prove useful in discussing holomorphic functions on spaces of distributions (chapter 5). Proposition 4.40
Banaah spaae.
Let E
=
t'"
1=1
E. where eaah Ei is a 1
On H(E), '0 and '8 define the same bounded
sets. Proof
Let F n
= In Ei for each positive integer nand i=l
let p be a T8 continuous semi-norm on H(E). without loss of generality that
We may assume
Holomorphic functions on Banach spaces
199
for every f
p (f)
n=o in H(E). integer n
n=o
We first claim that there exists a positive p
such that if f e: H (E) and f IF
Suppose otherwise.
n
= 0 then p (f)
o.
p
Then for every positive integer n we
I
can choose P , a homogeneous polynomial, such that P = 0 n n F n
~e now show that the sequence (P )00 I is n n= locally bounded. For each n let B denote the unit ball n of E • Hence there If x e:E then Xe: Fn for some integer n. n
exist Ai>o, i=l, ... n, such that II Pjllx+Ln A.B. ~M i=l 1 1
< co
for j=l, ... n.
By using the binomial expansion we can find An+l>o sue h t hat liP j fix + Ln + I A. B . ~ i=l 1 1
M + I
~n+l
for i = I , ... , n + I
and by proceeding in this manner, since each step only involves a finite number of polynomials, we can find a sequence of positive numbers, lip J· II x+/..,;,00 A B -< n=l n n
(An)~=l' such that
M+I for all j.
Hence {Pn}:=l is a locally bounded family of polynomials.
Since we only used the property P
I
n F
=
0
n
for each n it follows that {anPn}:=1 is also a locally bounded family of polynomials for any sequence of scalars for each n. is a locally bounded and hence a
To
bounded sequence of
200
Chapter 4
holomorphic functions. is T8 continuous this is impossible and establishes our claim. Let
(fa)aEA be an arbitrary TO bounded subset of H(E) and
let p be a
TO continuous semi-norm on H(E).
the proof we must show sup p(f ) a
a
L [(N) and we have already seen (example 2.47) that
i=l (H (C (N)) ,T) is complete.
Now suppose at least one Ei is
an infinite dimensional Banach space.
Without loss of
generality we may suppose El is infinite dimensional. '~n
denote the natural projection from E onto En'
n let q, n EE'n with II n 11= l .
Let
For each
By corollary 4.10 there exists
an entire function f on El with rf(o)
=
1.
201
Ho[omorphic functions on Banach spaces
tX>
Let g(x) Since gl
m
¢
n
(11
n
(x)) for all x in E. for every x in
ex) F
F
f(nlll (x))
n=2
m
= ~:l En and each compact subset of E is contained and
compact in some F
m
it follows that the partial sums of g We now show that
form a Cauchy sequence in (H(E) "0)'
g~H(E). Suppose otherwise.
Then there would exist a convex balanced
neighbourhood V of zero in E such that
Ilg Ilv = M
For
< '"
There
each positive integer n let Bn be the unit ball of En exists for each n a positive number 0
n
such that 0 B C:V. n n 4 0
Choose n a positive integer such that n > -
o X
£
--2lB 2
n
for wh i c h ¢ ( x ) f o.
By our construction there
n
If(y ) I ~ m
00
as m
----+
00
01 I 2Ym TBI and 2(--;;-)
Hence
+
I Ym "2(2x) = n + x
ny'
Since as m
I f(---..!'!.)¢ n n (x) I = I fey m) I ---+
00
this shows that g ~ H(E).
not complete.
and choose 1
£
V for all m.
I¢ n (x) I
----+
00
Hence (H(E) "0) is
By example 1.24 ((p(n E) , , ) and (l?(nE).B) o
are complete locally convex spaces for each positive integer n.
Hence H(E) is not T.S.'o complete.
H(E) is
not T. S.' 0, b complete and hence (H (E) "
a complete locally convex space. ,
0,
By corollary 3.34 0,
b) is not
By proposition 4.40
b = '0 and thus we have shown that (H(E),T) is not
complete for' = '0' the proof.
To,b"w'~,b
or '0'
This completes
Note that the above also shows that there exist '0
202
Chapter 4
oo
bounded subsets of H(I En) which are not locally bounded n=l whenever at least one En is an infinite dimensional Banach space.
We now briefly consider an extension problem which arises only in infinite dimensional analysis.
If F is a
subspace of a locally convex space E when can every holomorphic function on F be extended to a hOlomorphic function on E?
Two interesting distinct cases of this
problem arise when (b)
F is a closed subspace of E and
(a)
when F is a dense subspace of E.
Problem (a) concerns
an attempt to find a holomorphic Hahn-Banach theorem and will reappear in our discussion on holomorphic functions Example 4.42, which uses properties
on nuclear spaces.
of bounding sets, shows that in general we do not obtain a positive solution to this problem.
Problem (b)
is the
holomorphic analogue of finding the completion of a Exercises 1.89 and 2.94 are related
locally convex space. to problems
(a) and
Example 4.42
(b)
respectively.
This example is devoted to showing that
not every holomorphic function on c
o
can be extended to a
holomorphic function on loo' Let A =
(u )00 1 where u '" n n= n
(0, ... ,1,0 ... )
l'
for each
nth place A is a closed non-compact subset of
positive integer n.
c and of loo' By proposition 4.26 A is not a bounding o subset of Co and by theorem 4.31 A is a bounding subset of
Zoo'
Now suppose each holomorphic function of Co has a
loo'
holomorphic extension to f EH(c ) such that o IlfilA =
IIfl
k=
Ild
A
=
00.
By the above there exists If fE H( loo)
Zoo'
f then
Co
00 and this contradicts the fact
a bounding subset of
=
and fl
that A is
Hence we have shown that there
exist holomorphic functions on Co which cannot be extended holomorphically to
lQO'
203
Holomorphic functions on Banach spaces con~ex
If E is a locally
space with completion E then
E,E~,
thepe exists a subspace of
which is chapactepized by
the following ppopepties
(2) each holomopphic function on E can be extended to a holomopphic function on E 0 • /"
,A
(3 ) ifECFCE,
F a subspace of E, and each holomopphic
function on E can be extended to a holomopphic function on F then F c: E(.o' E~
is called the holomopphic completion of E.
Proposition 4.43
space then
El!)
If E is a metpizable locally convex
L-J
A
A is
whepe
the closupe of
ACE,A bounding
,..
A in E
U
Proof
A
E~
If !; E
then there exists
AC.E A bounding (!;n)nC E such that !;n lim f(!;n)
~
as n
--->- '!;
exists for every f
--->- co.
in H(E)
Since!;
E L9
E
and hence suplf(!;
n
n
)1
(nZ ) ,13)~ Zoo I
and hence deduce that (Hez ) "w) has the approximation l property. 4.75*
If E is a separable metrizable locally convex space
show that (H(U)"w) is quasi-complete for any open subset U
of E. Let E be a Banach space with a monotone basis
4.76*
(en)~=l
IIIk x n e n II -< II2:k+ I x n e n II for all k and all n=l n=l
i . e.
00
x =
2: x n e n n=l
E
E.
Let B be the open unit ball of E.
If
every automorphism of B has the form 00
2: n=l where
11
is a permutation of the positive integers,
for all nand sgP I"'nl
r
°
ensures that
A(P) A(P)
un = (0, ... ,0,1,0, ... )
nth position forms an absolute basis for the unit vector basis of Now let
E
Let
in
We refer to this basis as
A(P).
P = {(p(e ))00 } n n=l PEes (E) .
E
t
be a locally convex space with absolute basis
linear mapping from (en)~=l
A(P).
A(P)
is a complete locally
(un)~=l'
convex space and the sequence
is a
admits a continuous norm
if and only if there exists a continuous weight on consisting of positive numbers.
(Sn)~=l A(P)
E
into
A(P)
There is a natural
which takes the basis
onto the unit vector basis of
mapping is given by
A(P).
This
222
Chapter 5
and E
E
~(P).
is then linearly isomorphic with its image in ~(P)
is isomorphic to
if and only if it is a complete
locally convex space. When we identify a locally convex space containing an absolute basis with a subspace of a sequence space, we shall always assume that the above identification is used. Nuclear sequence spaces have a particularly nice and practical characterization as the following fundamental result shows.
This is the Grothendieck-Pietsch criterion.
Proposition 5.4
The sequence space
~
is nuclear' i f
(P)
and onLy if for' each (un):=l (an):=l E P ther'e exist a and an n. E P such that u a' for' :'i (a~)n=l
E
+
9,1
00
n n
n
~(P)
This criterion can be rephrased as follows:
nuclear' if and only i f for' each
(an):=l
sequence of non-negative r'eal number's, L:=l
i-
-
a
as
m
->-
00
Chapter 5
226
is a Schauder basis for the dual space in both
and cases.
Moreover,
~ and hence
II Til
[B]
is an absolute basis.
This completes the
proof. On combining lemma 5.6 and proposition 5.9, we immediately obtain the following result. Corollary 5.10
basis.
E~
If
E
Let
be a nuclear space with an absolute
then
A (P)
E'
B
~ A(P')
where
P'
Corollary 5.10 and the Grothendieck-Pietsch criterion may be used to decide when a given nuclear space with an absolute basis is a dual nuclear space.
We prefer, however,
to rely on
the following more practical criterion which covers most not all)
cases in which
E
(if
is nuclear, dual nuclear and has
an absolute basis. Definition 5.11
A locally convex space
nuclear space i f it has an absolute basis exists a sequence of positive real numbers
,00 1 Ln=l 8 n
1.
If
P
E
cs(E)
then A-nuclear-
228
Chapter 5
E.
defines a continuous semi-norm on
If
I~=l
x
x en E E n
let
I~=l
M
°
n Ix n Ip(e). n
Since m-l sup p(om(x - In=l xne )) n m
sup om m
~
I:=m
Ixnlp(e n )
M
is a of E. m-l x
Since e
->-
Now,
if
In=l
n n
-*
x
+
00
in
as
is a
q
n ->-
as
->-
00
and
continuous semi-norm on
'b
{r
'b-bounded subset
it follows that
00
m
0
and
a positive integer such that for every
g
in
H(U).
Hence
C . N
sup JCN(N)
J
and
finite
a z m
m
is J C N (N),
J
finite
T8
bounded.
N
245
Holomorphic functions on nuclear spaces with a basis
I mEJ,J
By lemma 3.28,
"
f~n~te
as
I mEJ,J f~n~te ..
if and only if
(H(U),T O)
a zm + f m
a zm + m
Iml =n
as n!
n.
for each non-negative integer
in
monomials form an absolute basis for
Since the
this shows
that they also form an unconditional basis for Since
To :;: Tw,b:;: To,b
this also shows that the monomials form
an unconditional basis for
(H(U),T
W,
If
b)
and
a z
I mEN (N) then
(H(U), TO)
fa
+
f
m
m
as
in
(H(U),T
a
+
00
0,
as
b)'
a+
oo
in
and hence,
since the monomials form a Schauder basis for a: + am
as
a +
for each
00
form a Schauder basis for (H(U),T
0,
b)'
If
associated with Let
on
p' (f)
T
T
U
and
is an open polydisc in a fully then
TO
is the barrelled topology
o
be the barrelled topology associated with
Since
H (U) .
is a
N (N)
This completes the proof.
Corollary 5.26
p
in
(H(U) ,TO)' (H(U) ,Tw,b)
nuclear space with a basis,
Proof
m
(H(U),T ) O and the monomials
is barrelled,
continuous semi-norm on
TO
H(U)
sup JCN(N)
J
is also a TO finite subset
is
TO
finite
continuous semi-norm and J
of
N(N)
p':;: p.
For each
the semi-norm
continuous and hence the set
V
=
{fEH(U);p(f)
(
I}
(and hence T) closed, convex, balanced and absorbing. is TO Thus V is a T-neighbourhood of zero and p' is T-continuous. Hence
T :;: To
and this completes the proof.
246
Chapter 5
Corollary 5.27 space
E
U
If
with a basis,
is an open polydisc in a fully nucleap then the following ape equivalent fop
op (aJ
(H(U),,)
is complete,
(bJ
(H(U),,)
is quasicomplete,
(cJ
(H(U),,)
is sequentially comple te,
(dJ
if
{ am}
and {"
a
LmEJ
is a set of complex numbeps
mE N(N)
m
Zm}
JCN (N), is a ,-Cauchy net,
I mEN (N) op (eJ
if
a zm m
,
E
J
finite
then
H (U) .
then the above ape equivalent to
w
{am}
and
I
(N) is a set of complex numbeps mEN m (N) p(amz ) < 00 fop evepy mEN
continuous semi-nopm on
I
(N)
amz
m
H(U)
then
E H(U).
mEN
We omit the proof since it is similar to the analogous result proved for holomorphic functions on balanced open sets in
ch~pter
3
(proposition 3.36).
Proposition 5.28 (H(E),d
If
E
is a peflexive A-nucleap space,
is an A-nucleap space fop
,=, o "
w
"
0,
b"
w,
b
then
op
'0·
Proof shown
is a reflexive A-nuclear space, is A-nuclear
then we have
(corollary 5.22) and
is A-nuclear (proposition 5.24).
By proposition 5.13,
the
247
H%morphic functions on nuclear spaces with a basis associated bornological topologies, A-nuclear topologies on
and
'0, b
, w, b'
are also
H(E).
There remains only the case,
'6
which we now discuss.
Our proof for this case can also be easily adapted to give direct proofs for the other topologies.
(on)~
6
Let
I
be
the sequence occurring in the definition of A-nuclearity. Since
E
is a complete A-nuclear space, the mapping:
H(E)
to
_+
H(E),
\"f(z)" f ( p ) ,
HeE)
is a linear isomorphism when
is endowed with the
'0
topology (it is also an isomorphism for any of the other is a
bounded subset of
is also a Ii CE)
.
If
P
is a
'6
bounded subset of
HCE)
continuous semi-norm on
TO
sup p«02)m aAz m) m A£B,md,(N)
H(E)
and
M
then sup Ae:B
I m£N(N)
omp(aAz m) m
~
M
I me:N(N)
1
O
oK
li=
where
There exist
)'.
(on)~=l
0 =
5.17, (oK)
m
O
such that
By lemma 5.18, we may choose a sequence of positive
H(U).
real numbers, ,00
(H(U),T
U
in
M u .
II b mw II (0 K) M
1
L
C
li=
mE: N(N)
om
-
V::J U
II Ilv)
-~s a regular
M
V open in E'S inductive limit, (dJ (e)
M
H(U ) '0
bounded linear functionals on
'0
continuous,
M
(fJ
H(U )
(g)
H(U )
then
is complete, H(U)
are
is quasi-complete,
M
is sequentially complete,
(a)(b)(c)=>(d)(e)(f)(g).
Furthermore,
if
is A-nuclear all of the above properties
E
U
are equivalent when Proof
=
E.
In any locally convex space
(d)=>(f)=>(g).
M
Since
easily show that
H(U )
(g)=>(d).
(a)=>(b),
and
(a)=>(e)=>
has an absolute basis, Now suppose
(b)
holds.
be a semi-norm on
H(U)
subsets of
By proposition 5.25, we may suppose
H(U).
which is bounded on
one can Let
p
bounded
'0
a zm) sup p( 1 Lme:J m JCN (N) J finite
for every
I me:N (N) Let
V = {fe:H(U) ;p(f)
absorbs every
continuous.
l}.
Then
H (U) .
V
bounded subset of
'0
easily seen to be this shows that
~
in
'0
V
is convex, balanced and H(U).
closed and
Since
V
is
is infrabarrelled
is a neighbourhood of zero and hence
Thus
(b)=>(a) .
(b)
and
(c)
p
is
are equivalent
254
Chapter 5
by theorem 5.29, since a locally convex space
F
is infra-
barrelled if and only if the equicontinuous subsets and the strongly bounded subsets of M H(U )
Now suppose
coincide.
F' S
By propositions 5.9,5.21,
is complete.
and 5.25, the monomials form an absolute basis for both (H(U),TO)S
and
(H(U),To,b)S
If
T E (H(U),To,b)'
partial sums in the monomial expansion of
T
in (H(U),T )' and hence TE (H(U),T )" O O this completes the proof for arbitrary U. Now suppose
E
then the
form a Cauchy net Thus
is an A-nuclear space and
(d) => (e)
U = E.
and
By
proposition 5.28, the monomials form an absolute basis for both (H(E),T) and (H(E),T b)' By lemma 5.1, T = T if and o 0, 0 o,b only if (H(E),T O)' = (H(E)'To,b)' and hence (e)=>(a). This completes the proof. Corollary 5.31
basis. on
H(U)
Proof
Then
Let
be a fully nuclear space with a
E
on
TO = To,b
i f and only i f
H(E)
for every open polydisc By corollary 5.30,
U =
T
TO = To,b
in E.
T
o o,b is a regular inductive limit.
on
if and only
H(E)
Since the space of if H(OE') S germs about any compact polydisc is regular if and only if the space of germs at the origin is also regular,
a further applic-
ation of corollary 5.30 completes the proof. Corollary 5.32
If
U
is an open polydisc bn a Frechet
nuclear space with a basis, only i f
M
H(U )
Example 5.33
( a)
If
admit a continuous norm,
2.52).
then
E
on
T6
H(U)
i f and
then
is a Frechet space which does not TO F To
Hence, by corollary 5.31,
has a basis then particular,
TO
is a regular inductive limit.
H(OE')
H(O (N))
if
on E
H(E),
(example
is also nuclear and
is not a regular inductive limit.
Sis not a regular inductive limit since
I[
does not admit a continuous norm.
We have already proved
In
255
Ho[omorphic functions on nuclear spaces with a basis this directly in example 3.47. that
H(OE)
a :'f)JYL
More generally, the above shows
is not a complete inductive limit whenever
space with a basis and
E' S
E
is
does not admit a continuous
norm. (b) of
E
If
E
is a Frechet space and
K
is a compact subset
then lim
H(K)
(HOO(V), II
IIV)
--+
V::>K
V open (proposition 2.55)
is a regular inductive limit corollary 5.30, whenever
U
since
E
is a k-space,
is an open polydisc in a
'0
~.1h.
and hence, by
'a
=
on
H(U)
space with a basis.
This is a particular case of the result proved directly in example 2.47. We now characterize the Borel transform of functionals.
'w
analytic
This characterization was originally used to
prove the topological isomorphism of theorem 5.29, and leads to a simple criterion for comparing U
and
'0
'won
when
is an open polydisc in a fully nuclear space with a basis.
Proposition 5.34
U
Let
be an open polydisc in a fully
nuclear space with a basis. Moreover, a subset
of
V
only i f the germs in
(H(U)"w)'
in
have
Let U
T e:
are defined and uniformly bounded
(H(U) "w)'.
IT(f) I ~ c(V) Ilfllv T
and
for all V.
UM.
There exists a compact polydisc
such that for every open polydisc
depends only on
is a M
HHY(U).
is equicontinuous i f and
(H(U)"w)'
B(V)
onto
on the compact subsets of some neighbourhood of Proof
'" B,
The Borel transform,
vector space isomorphism from
K
H(U),
f
in
V,
KC VC U,
where
H(U)
Moreover, the set of all
we c(V)
T
which
satisfies the above inequalities forms an equicontinuous subset By lemma 5.18, we can choose for each neighbourhood
a
V
= (on):=l'
of
K
on >1
a sequence of positive real numbers for all
nand
I n=l oo
1 on
(b)=>(c)
B.
lemma 5.1 shows that
(d) are equivalent.
(c)=>(a).
By proposition 5.34
(~)
Since
= HHY(V)
open polydisc in a fully nuclear space with a'basis 5.23),
(d)
where
V
is satisfied and
E
is an open polydisc in
a
m
Z
JCN eN) ,J
is a To-bounded subset of open polydisc
sup
V,
I g(z) I
S'
then
m }
finite
H(V)
and hence is locally bounded.
K
Hence for each compact polydisc
Hence
(corollary
and (e) are equivalent.
(f)
If
for any
KC VC U,
(f) and
subsets of
and
completes the proof. Example 5.36
(a)
If
U
is an open polydisc in a Frechet
nuclear space with a basis or in a J)111.
space with a basis,
then
This result follows from corollary TO = T W on H (U) . We 5.35 since condition (d) is easily seen to be satisfied.
have already proved this result for arbitrary open subsets of ~
JYL
spaces
(example 2.47)
Fr~chet nuclear spaces
TO
we showed that
r
TW
and for entire functions on
(corollary 3.54). on
(b)
H((N x [(N)).
In example l.39,
This is a particular
case of the following result which is an immediate consequence of corollary 5.35 and example 1.23.
E
If a basis,
is an infinite dimensional fully nuclear space with
then
T
W
on
We also obtain a topological characterization of
(H(U),TW)S
in certain situations.
This is illustrated by the
following proposition.
Let
Proposition 5.37
U
nuclear space with a basis
be an open polydisc in a fully E.
The following are equivalent:
(a)
(b) (c)
(H(U),Tw)S
V open has the monomials as an absolute basis
and
(H(U),T)
the
T
w
w bounded.
Moreover,
if
E
is semi-reflexive,
bounded subsets of
is an A-nuclear space,
equivalent to the following: (d)
(H(U).T )
is semi-reflexive,
(H(U),T )
is quasi-complete.
W
(e)
W
H(U)
are locally
then the above are
Chapter 5
260
The monomials form an absolute basis for
Proof
theorem 5.21.
HHy(U M),
By theorem 5.29,
B
be a K
in
,
CUM)
by
may be identified, via the Borel transform, with
are equivalent.
(b)
HY
lemma 5.17 and corollary 5.23,
An appl ieation of lemma 5.1 now shows that
H (U).
H
Now suppose
H(U).
sup A
and Let
For each compact polydisc
there exists an open polydisc
U
Ca)
is satisfied.
I mEN eN)
bounded subset of
w
ee)
II 'LeN) mEN
M
aAzml1 m
W
such that
< "".
K+W
Choose a sequence of positive real numbers 0 = (6n)~=1' ,"" 1 for all nand Ln=l ~ < 00, and V an open po1ydisc in n
such that
:;
:;
6(K+V) C K+W.
I mEN(N)
I mEN(N)
om
1 om
Hence
sup AE[
.
M
I a mA I . II z m Ilo( K+V) < ""
and Bt
is a locally bounded and hence a Since
I mE:N.(N)
suplaAb I AE:r m m
sup
I mE: NCN)
bounded subset of
H (U) •
261
Ho!omorphic functions on nuclear spaces with a basis
for any set of scalars
(N) this proves that mEN has the monomials as an absolute basis. If
(H(U),Tw)S
{bm}
then there exists a of H (U) such that I cp (w m) I = I ami ::; N (N). Hence
cP E
((H(U) ,Tw)S)'
where
I
It now follows that 1 i es in
is semi-reflexive and Suppose be a a
m
T
(b)
B
B
for every
m
in
cp
and hence
H (U).
E
Hence
(c) => (b).
is satisfied.
bounded subset of
w
supi a~1
\I wml!
supla~lzm,
mE N I.E r and we have shown
H(U)
bounded subset
{I mEN (N)
B
(N)
Tw
Let
H(U).
C =
{I
aAz m} m AEr N(N) let
(N)
mEN For each m in
and let
A
1/J
(\ L
-:lEN
(N)
b
wm)
m
for every and the monomials form an absolute basis for
(H(U),TW)B'
1/J
continuous form on
is a
(H(U),T)'. By semi-reflexivity w 8 identified with an element of H(U), that is,
Hence, for every compact polydisc open polydisc
W containing
K
K
If
H(U) E
and hence
there exists an
B
sup l\aAzml!w < 00. A m is a locally bounded
(b)=>(c).
is an A-nuclear space then
by proposition 5.24, and hence basis by proposition 5.9.
U,
W
may be
such that
By nuclearity, it now follows that subset of
in
T
1/J
Hence
(H(U),T) w
(H(U),TW)S (b)
and
is nuclear
has an absolute (d)
are equivalent.
262
Chapter 5
In general,
it is easily seen that
satisfied then
If
(c)=>(e).
is
(e)
is a quasicomplete nuclear space and
hence it is semi-reflexive.
(e) 0;} (d)
Thus
and this completes
the proof. The Borel transform of
'I)
analytic functionals is
treated in exercise 5.81,
§5.3
HOLOMORPHIC FUNCTIONS ON
DN
SPACES WITH A BASIS
Using the results of the preceding
section and modificat-
'w =
ions of the techniques used to show
on
'6
is a Banach space with an unconditional basis show that nuclear
'0
DN
=
'I)
on
H(U)
when
U
H(E)
when
E
(section 4.3), we
is an open polydisc in a
space with a basis.
We begin by recalling some fundamental
facts about
DN
spaces.
s,
the space of rapidly decreasing sequences,
is the
Frechet nuclear space with a basis consisting of all sequences, of complex numbers such that
is finite for all positive integers
m.
generated by the norms
is a universal generator
s
The topology of
for the collection of nuclear locally convex spaces, locally convex space ~omorphic
E
i.e.,
s
is a
is nuclear if and only if it is
to a subspace of
s
A
for some indexing set
A.
A
depends on the cardinality of a fundamental neighbourhood system at the origin in
E.
In particular, any Frechet nuclear
space is isomorphic to a closed subspace of Let
Definition 5.38
E
be a metrizable locally convex
space with generating fami ly of semi-norms for all
n.
E
is a
DN
N
s .
(Pn) ~=l'
Pn:; Pn+l
(dominated norm) space i f there is a
263
Ho[omorphic functions on nuclear spaces with a basis
continuous norm
such that for any positive integer
a positive integer
there exist
k
E
on
p
nand
such that
C>O
for an
r>
The fundamental result concerning nuclear
o. DN
spaces is
the following proposition.
A metrizable nuclear locally convex space
Proposition 5.39
is a of
space i f and only i f it is isomorphic to a subspace
DN
s.
E
Now let
is isomorphic to w
m
(wm,n)~=l
be a Frechet nuclear space with a basis. A(P)
where we may suppose
P = (wm):=l
for all
for all
E
m
and
n
and w
w
m
Ln,w
for all
m,n
m,n
"f0
O
and
k
2
is isomorphic to
(i) (ii)
if
and
(b)
m
for all
and
m
n
wm,n
Sm,n
~
all
I
p 00 (wm,n(Sm,n) )n=l positive integers (a)
where P = (wm):=l' and the following m
for all W m+l,n
Sm,n
then
A(P)
for all
> 0 W m,n
n,
for all
I ,n wk,n
wm = (w m , n)~=l hold:
Proof
Il.
P = (wm):=l' is isomorphic to where E A(P) all m and for each W = (wm,n)~=l for m positive integer m there exist a positive
integer
(e)
and
m
(;
m
for any
[P] m
and
n
and
and
p.
We
are equivalent by proposition 5.39.
do not prove the equivalence of (b),
(c)
and
(d) here.
See
the not es and remarks at the end of this chapter for a reference. (c)=>(e). and
n
Since
(wm+l , n)
2
::: Wm,n Wm+2,n
for all
m
we have W m+l,n
:;:
W m,n W m+l,n W m,n
W m+2,n
and hence
W m+l,n
:::
W m+j+l,n W m+j,n
for all positive integers m, nand
j.
Chapter 5
266
Hence
W
Wm, n ( and
m+p,n
(e)=>(d).
Wm+l,n )p~ W m,n
W
p
)
Wm+j+l,n
j=O
W
.
m+J,n
(c)9(e).
We first prove by induction on
(WI ,n (
n
m,n
00
)n=l
E
m,
assuming
for all positive integers
[P]
(e), that and
p
m.
Wl,n The case
m=l,
p
arbitrary is trivial.
is true for the positive integer
m
induction hypothesis there exist
Cl>O
Now suppose the above
and for all
p.
By our
and a positive integer
such that
J ,n
By condition
(e)
there exist
C
2
> 0
n.
for all
W.
and a positive integer
such that
n.
for all Hence WI ,n (
where Thus
:::
C
C
Ic 1 . C.2
C
l
2
w. C J ,n wk,n ::: and
W
2p )
WI ,n (
W 2" ~)r-
wI, n
m,n
2 W Q"n
j +k.
Q,
W m+l,n )p ::: C W
WI ,n (
2
Wm+l,n
~,n
for all
n
and
wI,n W
(WI ,n
If we let
(
~
wI • n p=2
1)
00
)')
E
[P].
n=l we obtain (d)
completes the proof.
and hence
(e)=> (d).
This
k
267
Holomorphic junctions on nuclear spaces with a basis Condition (e) of proposition 5.40 arose in our study of holomorphic functions on Fr~chet nuclear spaces with a basis
and is the only one of the above equivalent conditions that we shall use from now on.
In the original papers on holomorphic
functions on nuclear spaces, a Frechet nuclear space which satisfied condition (e) was known as a B-nuclear space.
DN
relationship between B-nuclear spaces and
The
spaces with a
basis was noticed afterwards. Example 5.41
(a)
Let
~ (an):~l
a
be a strictly increas,00 qa n < 00
ing sequence of positive real numbers such that q, O-
lim inf j
---+ '"
j
1m. I J
1
/lm.1 J
) k,
,
s. ~ im I (8 k n) J
/
J+OO
e
'
I
111m.J I >-
e'
272
ChapterS
Since
C(k)8\m\
for all
mE N(N),
we must have
)
lim sup j
----+
00
This is a contradiction,
since
c5' > 1
is a strictly increasing sequence. Let
m. z J
fez)
Since each monomial is continuous and theor~m
2.28 implies that
f
a = (an)~=l
n
E.
let
'" an
Choose
= anun ~
is a Frechet space,
is an entire function if the
above series converges at all points of Let
E
E.
be an arbitrary element of
where
(un)~=l
E.
For each
is the unit vector basis of
a positive integer such that 1 2 ~ all n>-~. For each j let m , mj , ... , mj j coordinate of m. E N(N) and let J
...v
an E Vk+l
for
be the first
~
273
Holornorphic functions on nuclear spaces with a basis
for
i
1, ... , )(, .
We have m. a
J
where
for all
such that
j>J(,
(the terms between
J(,
and
j+n k
are also less than one but we need a sharper estimate). Now given any positive integer and hence nk+j
> )(,1:
law
n k ,n
(Sk
,TI
)PI~l
p,
for all
in particular for all
j
[Pl n
~
)(,1 > J(,.
sufficiently large, we
have
s.
p-l
J
1
1
and thus
lim sup j
Ia
m. J
1
II m.J I
I
-->-00
lim [ j
+00
1
Hence if
1/ lmj1 jP-1
Chapter 5
274 where
lim sup j
if
--+=
o 1:;: i ~
for mi
Since all
j
::;
1 ffij
c
i
:;: 1
(note
0
0
= 1)
~.
for all
1
if
i
we have
and
0 :ir.::; ].
for
Hence
i.
lim sup j
rM, c~
.
1
As
is greater than
w
zero and hence
f
€
1
and
p
is arbitrary, the limit is
H(E).
Hence m.
z J
r'
'"f (z.)
j =1 (, j
and establishes our claim. Since
T
'6
is
(nVl)~=l
continuous and
countable open cover of
there exist
E
is an increasing
c (1)
> 0
and
01
a
positive integer such that
for every
f
H(E).
in
In particular
m £ N (N)
for all
h were
(on)~=2
Let
TIl = 1 .
be a sequence of positive real numbers,
IT a = a is finite. By the n=l n above, we can choose inductively a strictly increasing sequence
such that
(nk)~=l'
of positive integers, positive numbers
(c(k))~=l
for all
mEN (N)
and all
Let
K
aU (n ):=l'
If
V
i
K
n
= 1, l such that
and a sequence of
k.
is a compact po1ydisc in
is any neighbourhood of
K
then we can choose, by
lemm" 5.18, a sequence of real numbers, £
n
> 1
all
nand
,'" 1 < '" Ln=l' £
n
and
E.
E W
(£n)~=l
with
a neighbourhood of
Chapter 5
276 E (K+W) C
zero such that
K
V.
Since
K+W
k
there exists a positive integer
Hence, for any IIT(f)11
such that
f E H(E),
~
I
~
I
(proposition 5.25)
(N)IT(zm)11 mEN
mEN
(N) c (k)
c (k )
Since
is a neighbourhood of
II
.
fll
. II
V .
zmll
\'
K+W
I mEN(N)
L
m E
V was arbitrary, this shows that
compact subset
K
of
E.
Hence
T
c(k) II mil (N) --m- z (K+W) mE N E E
T
is,
is ported by the w
continuous and
this completes the proof. Theorem 5.24 immediately leads to a strengthening of some of our earlier results. (a)
If
particular if
E
The following are now easily verified.
is a nuclear E
=
s
or
H([))
DN
space with a basis (in then
(H(E)"o)
is a reflex-
ive A-nuclear space. (b)
If
U
is an open polydisc in a nuclear
with a basis, then
(H(U)"o)
DN
space
is a fully nuclear space with a
basis. (c)
If
E
with a basis then
is the strong dual of a nuclear H(OE)
=
lim
(Hoo(V), II
V;)O,V open
IIV)
DN
space
is a complete
277
Holomorphic functions on nuclear spaces with a basis
regular inductive limit. Thus we have examples of non-metrizable locally convex spaces in which the space of germs about the origin is complete and regular. In chapter 6, we prove, using tensor products and a result of Grothendieck,
If on
§5.4
E
H(E)
the following converse to theorem 5.42.
is a Frechet nuclear space with a basis and then
E
is a
DN
T
o
=T
0
then
x
where
e(~
M.
I
)
is positive since
Hence, by choosing
c
x
c ~ , V ~,
h(j~'+l)
~
x
cV ~ , + 1
, + 1.
sufficiently small and positive, we
have
::
Since jn a: z n
n
the same estimate also holds for all Thus we can choose a sequence of positive real
numbers sup n
M.
(nE) .
If
n
co
PIF = Ii=l(~i) where for some neighbourhood
~. E 1
F'
V of 0 I: =1 II tjJ ill ~ < then, by the Hahn-Banach theorem, there exists a neigh-
and
of zero in
W
Ilrill ~
m 1 i m \' Li=l m+oo
P
, 2.
is a balanced open subset of a Frechet nuclear
space show that 5.78*
is a nuclear space
= HHy(E S ).
((H(U)"o)S )S ';' (H(U)"o)· is a compact subset of a fully nuclear space oo is a regular inductive lim (H (V) , II ---+
V'::>K,V open
limit, show that space.
H(K)
is a quasi-complete locally convex
292
Chapter 5
5.79*
Show that a fully nuclear space with a basis is
ultrabornological.
5.80
U
Let
with a basis.
be an open polydisc in a fully nuclear space {b m }
If
is a set of complex numbers,
(N)
mEN show that there exists a '0
analytic functional T on mEN (N) T(zm) = b such that if and only if for all m m E 00 < for every H (U) • I N(N) lambml I (N) am z mE mEN
5.81*
Let
E
If
{b m }
with
b
5.82*
for all
analytic functional on
if and only if each m J of N(N) infinite subset contains an infinite subset J' m such that I mE: J,b mw E H(OE')· S E
T(Zm)
and suppose
is a set of complex
(N)
mEN show that there exists a '0
numbers,
spac~
be a reflexive A-nuclear
is complete.
U
mE N(N)
be a nuclear power series of type 1.
a = (an):=l. for all
n.
(m, n)'
Suppose each
an
is an integer and
For any positive integers
(rna
man
m
and
n
a
n+l let
Let a +n n
>
,0,
t...- - - - - man and for
position
let rna
rna
n
if
(m,n)'
b. J
otherwise. Show that there exists a such that is not
5.83*
'I
'0
analytic functional on
T(zj)
o
= b. for every J continuous.
in
For each non-negative integer
N(N).
n
E.
Let
T
let
denote the vector space of continuous alternating forms on the locally convex space
AI(a)
Show that
n
linear
293
Holomorphic functions on nuclear spaces with a basis
{CPn)~=o;PnS LACnE),L~=oIIPnIIKn= PKC{Pn}~=o)
< co}
and endow
HACE)
with the topology generated by the semi-
norms
as
ranges over the compact subsets of
E
PK
K
is a fully nuclear space with a basis,
a closed complemented subspace of Hence deduce that E
is a nuclear
5.84*
If
E
DN
show that
CHCE)"o).
is a reflexive nuclear space if
HACE)
space with a basis.
is a Banach space with a Schauder basis,
that the monomials of degree
n
show
form a Schauder basis for
C{PCnE),,), n=I,2,... If, in addition, E has the o Dunford-Pettis property and its basis is shrinking, show that they also form a basis for
§S.6
NOTES AND REMARKS Although D. Hilbert
[332]
first suggested,in 1909, a
monomial expansion approach to holomorphic functions in infinitely many variables, this idea was not developed until recently and most of the results presented in this chapter were discovered within the last four years.
Indeed, many of
the original research articles are still only available in preprint form. Nuclear mappings and nuclear spaces, as well as most of their fundamental properties, are due to A. Grothendieck
[28n.
Further accounts of the linear theory are given in the very readable book of A. in the notes of Y.
c.
Pietsch [570], Wong
[717].
this chapter is given in S.
in P.
Kree
[403,404], and
A survey of the resul ts of
Dineen [198],
from a slightly different point of view,
and a full account, can be found in
S. Dineen [197]. Lemma 5.1 is due to P.J.
Boland and S. Dineen [91].
Apart from its uses in [91], we have used it in this chapter to shorten a number of the original proofs.
Proposition 5.4
294
Chapter 5
is the classical characterization of nuclear sequence spaces due to A.
Grothendieck
[287]
and a proof is given in A.
Lemma 5.6, corollaries 5.7 and 5.S,
[570].
Pietsch
proposition 5.9 and
corollary 5.10 are all probably known to research workers in nuclear space theory, but we have been unable to locate an exact reference. [202]
A-nuclear spaces were introduced by S.
as a tool in studying holomorphic functions
of independent interest. [202] P.J.
Dineen
but may be
Proposition 5.12 is due to S.
Dineen
while proposition 5.13 and corollary 5.14 are given in Boland and S. Dineen [91]. Fully nuclear spaces and fully nuclear spaces with a basis
were introduced by P.J.
Boland and S. Dineen [90]
solve the basis problem for space with a basis.
(H(E), '0)
This article,
[90],
when
E
in order to is a nuclear
contained the motiva-
tion for many of the results in this chapter,
introduced the
concept of multiplicative polar (definition 5.16) and contains the key results,
5.17 and 5.1S.
Lemma 5.19 is due to S.
Dineen
[202] . Theorem 5.21, corollaries 5.22,5.23 and proposition 5.24 are due to P.J.
Boland and S.
Dineen
[90], and S. Dineen [202]'
the latter containing the results on A-nuclearity. using a proof similar to that of theorem 5.21, A.
whenever
E
Benndorf [56)
admits a finite dimensional Schauder
has shown that decomposition
Recently,
(H(E)"O) (i.e. dim(E ) < for each n in definition 3.7) n is a Frechet-Schwartz space with a finite dimen00
sional Schauder decomposition.
Propositions 5.25,5.28 and
corollaries 5.26,5.27 are given in S. Dineen 5.29 is due to P. J.
[202).
Boland and S. Dineen [90].
Theorem
The original
proof of this proposition, which used proposition 5.34, has been shortened here by using lemma 5.1.
Corollaries 5.30,5.31,
5.32 and example 5.33 may be found in P.J.
Boland and S. Dineen
(91], and S. Dineen [202], while proposition 5.34 is due to P.J.
Boland and S. Dineen
[90].
Corollary 5.35 and example
5.36, both of which involve an application of lemma 5.1, are due to P.J.
Boland and S. Dineen [91].
Proposition 5.37 is a
slightly improved version of a result proved in P.J.
Boland and
295
Ho[omorphic functions on nuclear spaces with a basis S.
Dineen [91]. In his investigations on the mathematical foundations of
quantum field theory with infinitely many degrees of freedom, P.
Kree [407,408,409,410] used the nuclearity result of P.J.
Boland
[86] and L. Waelbroeck [713]
quently he wished to know if locally convex space.
This,
o
was a bornological
)
together with the results of
further questions of Kr~e concerning
sections 5.1 and 5.2, H (~)
(theorem 3.64) and subse-
(H(s),T
and the possibility of a kernels theorem for analytic
functionals in infinitely many variables
(see chapter 6)
motivated much of the research described in sections 5.3 and 5.4. To solve these problems, S. Dineen [195,202],
introduced
the concept of B-nuclear space and proved theorem 5.42. Subsequently,
D. Vogt,
in a private communication, showed that
B-nuclear spaces and nuclear DN-spaces with a basis coincided Cproposition 5.40).
DN
spaces are due to D. Vogt and have
played an important role in the development of structure theorems for Frechet nuclear spaces.
We refer to D. Vogt
[703]
for an excellent survey article on nuclear DN spaces and to E.
Dubinsky [212]
for a comprehensive account of recent devel-
opments in nuclear Frechet space theory. ences for proposition 5.39 are D. Vogt [211].
The original refer-
[702]
Proposition 5.40 is due to D. Vogt
Lemma 5.43 is due to S.
Dineen
[199].
and E.
Dubinsky
[702,703]. The particular
case of proposition 5.44 described in example 5.45 is due to P . J.
B0 land and S . Dineen [ 9 2], whi 1 e the general res u 1 t
appears in S. Dineen [198,199]. HC£))
is given in P.J.
An earlier partial result for
Boland and S.
[704] and M. Valdivia [690]
have
Dineen [91].
prove~
deep result quoted in example 5.45,
i.e.
independently, the ;DC,I)
~ sCN).
Example 5.46 follows from results proved in P.J. S.
D. Vogt
Boland and
Dineen [91]. The remaining results in section 5.4 are to be found in
296
S.
Chapter 5
Dineen
[199], where the connection between extension
theorems for holomorphic functions on subspaces and topological properties of
H(E)
is established.
Corollary 5.50 (the
holomorphic Hahn-Banach theorem) was first proved by P.J. Boland [83] and B.
using a direct approach.
In [161], J.F. Colombeau
Perrot generalise this corollary by showing that a
Fr:chet-Schwartz valued entire function on a closed nuclear subspace of a JJJ~ space, to
E.
E,
can be extended holomorphically
An extension theorem for nuclear holomorphic functions
on Banach spaces is given in R. Aron and P.
Berner [26]
and
using this theorem and a uniform factorization theorem for holomorphic functions on :tJ:; J.F.
rrz
spaces
Colombeau and J. Mujica [156]
of corollary 5.50.
(see exercise 2.105),
give an alternative proof
A further proof of this corollary and also
of theorem 5.48, using the symmetric tensor algebra
(definition [48~.
6.54), has recently been obtained by R. Meise and D. Vogt In [488],
R. Meise and D. Vogt show that the holomorphic Hahn-
Banach theorem is not valid for certain nuclear Frechet spaces. A different kind of extension result for entire functions on a nuclear subspace of a locally convex space is given in A. Martineau [453]. In chapter 6,
section 4, we prove a number of structure
theorems for holomorphic functions on infinite type power series spaces. The role of nuclearity in infinite dimensional holomorphy is much more extensive than that outlined in this book. is mainly due to our choice of topics.
This
In Appendix I, we see
that it appears in the study of convolution operators on spaces of holomorphic functions, ""D
in solving the Levi problem and the
problem and in infinite dimensional holomorphic sheaf theory.
Apart from these topics we also find nuclearity appearing in A. Martineau's study [453] als in several variables, continuation
[712]
J
of the supports of analytic functiofr in L. Waelbroeck's result on analytic
in B.
Kramm's
[398,399,400]
interesting
classification theorems for Fr{chet nuclear algebras, analytic spaces and Stein algebras and in M.
~ ~~
Schottenloher
[~61
Chapter 6
GERMS, SURJECTIVE LIMITS, € -PRODUCTS AND POWER SERIES SPACES
The last two chapters dealt with scalar-valued holomorphic functions defined on special domains in special spaces. return in this chapter to the general theory, further methods -
the
'rr
topology,
We
and present three
surjective limits,
and
[-products - for studying the relationship between the topologies
'o"w
and
'0·
The
'rr
topology aims at removing
geometric restrictions on the domain,
surjective limits are
used to generate spaces of holomorphic interest and using
E -products
we study vector-valued functions.
Apart from the problem of the different topologies we also discuss in this chapter other problems of general
interest such
as the representation of analytic functionals and the completeness of
The final
H(K).
section of this chapter is devoted to
holomorphic functions on the strong duals of certain power In this case,
series spaces.
the results of chapter five are
combined with some interesting estimates to obtain a number of representation theorems.
§6.1
HOLOMORPHIC GERMS ON COMPACT SETS In chapters 2,3 and 5 we obtained various results
concerning the regularity and completeness of compact subset of a locally convex space.
H(K),
K
a
The positive and
negative results we obtained show that these are indeed complex questions and not unrelated to one another. assumed that condition -
K e.g.
In most cases,
we
satisfied a rather restrictive geometric that
K
consisted of a single point or was a
298
Chapter 6
balanced set or a polydisc. internal structure of
K
This effectively meant that the
played no part in our investigations
and that essentially we were studying local phenomena.
Here we
look at the global theory by considering arbitrary compact sets. We do not concern ourselves with the regularity and completeness of
H(O)
- as we have considered these questions in chap-
ters 3 and 5 and shall give a further example in the next section - but look at
H(K)
where
is a compact subset of a
K
Frechet space or of a fully nuclear space with a basis.
The
method of proof used for metrizable spaces was motivated by the proof of proposition 3.40 and this,
in turn,
provided the
motivation for the fully nuclear space case. We have already seen in proposition 2.55, that
K
regular when
H(K)
is
is a compact subset of a Frechet space and
K is balanced.
that it is complete when
We now extend the
latter result to arbitrary compact sets. Theorem 6.1
then
E
is a compact subset of a Frechet space
is complete.
H(K)
Proof
K
If
Since
H(K)
is a
that it is quasicomplete. topology on
H(K).
Let
DF Let
T
space, Tl
it suffices to show
denote the inductive limit
be the locally convex topology on
generated by all semi-norms which have either of the
H (K)
following forms: (* )
n!
where
p
is a
Tl
continuous semi-norm on
H(O),
P2(f)
(xn)~=l'
where and
(Y~)~=l
nand
(k)oo
(x~)~=l
are two sequences in
are null sequences in
n n=l
E,
K,
(Yn)~=l'
xn+Yn = x~+y~
for all
is a strictly increasing sequence of positive
Germs, surjective limits,
E
299
-products and power series spaces
integers. By proposition 2.56, subsets of
H(K}
and
T
We now show that (gS}SEB n
be a
T
T
and
TI
TI .
~
(H(K},T)
is quasi-complete.
bounded Cauchy net in
is a positive integer, then using
of the form (*), net in (If' (nE},T
define the same bounded
"n
w
If
H(K).
T
Let XEK
and
continuous semi-norms
we see that (d gS(xYn!}SEB is a Cauchy )' Since (lP (nE},T w ) is complete ~(nE),
(corollary 3.42), there exists an element of
Pn,x'
such that as As
(gS}SEB
zero in
E
Band
and
such that
lip n, x II IV
~
4 let
for all
n
y
and
y,y' E W satisfy
W
arbitrary.
,00
M
Ln -
- nI
4
~
n
E.
n
l For any n
l
Ln=o Since exists
qn
I
n
x+y
and
L~=o
f (x) (y)
Choose
=
x'+y'
x.
~
in
E
This implies
n.
For any
x
P (y) . Let n,x and let E > 0
in
x,x'
be
a positive integer such that f E H(K)
let n
1\
dnf(x} nl
(y) -
l
Ln=o
"dnf(x') n!
is a T continuous semi-norm on B E B such that 0
qn (gs -gs ) I I 2
W of
for all
gs E H(K+4W)
and all non-negative
M
and
in
00
Hence
XEK
S E B,
->-
is bounded there exist a neighbourhood M> 0
supllgsllK+4W SEB
for all that
S
for all
S I' S 2
~
So'
H(K}
(y' )
I
there
K E K
300
Chapter 6
n
Pn,x(y)
l
Ln=o Pn,x' (y')
-
E
and
Thus, and
there exists an f(x) (y)
f
= f(x+y)
in
Hoo(K+W)
for all
x
in
such that
II fll
K+W
K
in
W.
Due to the form of the semi-norms immediate that (H(K) ,,)
ge
-+
f
as
S
is quasi-complete.
ogy associated with
,
barrelled space and
,(. '1
hence
-+
(H(K)"l)
on
and
This shows that
in
00
Let H(K).
and
(*)
(H (K) ,1:).
(. M
it is
(**)
Hence
be the barrelled topol-
Since
is a '2:S ' I
and
have the same bounded sets.
is a barrelled
over, by proposition 3.6, hence complete.
y
it follows that
(H(K)"2)
(H(K)"2)
'2
and
DF
space.
More-
is quasi-complete and
We complete the proof by showing that
'1
'T
The situation now is rather similar to that of proposition 3.40,
and an examination of the proof of that proposition
shows that we only need find a fundamental system of bounded subsets of
H(K)
(B)oo such that ,k A B is , n n=l' Ln=l n n 2 closed for any finite sequence of numbers (An)~=l' in order >
,
to complete the proof. (Vn)~=l
Let
be a decreasing fundamental neighbourhood
E
system at the origin in sets and for each
n
let
consisting of convex balanced open Bn
be the closed unit ball of
H= CK+V n) . k
Let y E r
-+
h y
00.
'(, n
E B
n'
Ln=l AnBn
hE H(K)
-+
By corollary 3.39,
(H(K+Vn)"o)' h
E
For each
y
in
By using subnets,
Bn
r
in the
,
topology as
is a compact subset of let
hy =
L~=lAnhy.n
where
if necessary. we see that there
Germs, surjective limits,
exists a
rv ->-
h
in
E
Hence
n.
closed subset of
->-
nl
nl
B
B
H(K)
of
H(K) V
for
some neighbourhood of
K,
and '2 '
an arbit-
K
(i.e.
Hoo(V),
if and only if the elements
B
K
and the local
is coherent in
if and only if there exists a
W of zero such that (y)
f
K
is contained and bounded in of
Taylor series development of elements of
for every
K
,
satisfy uniform Cauchy estimates over
neighbourhood
00
of a locally convex space.
for some neighbourhood of
in
k , and hence is a In=IAnBn This completes the proof.
We now look at the regularity of
A subset
x
for all
and
H (K) .
rary compact subset
->-
/I
dnh(x)
h
y
00.
'" dnh(x)
'"h
as
n
Hence
y
/'0
This implies that
h
->-
y,n
\
as
(H(K+Vk)"o)
h
k Ln=l An B n and
~
all
such that
h
k h = \Ln= IA n h n
301
-products and power series spaces
E Bn' n=l, ... ,k, n (H(K+Vn)"o) for all n.
in
hy
€
in
B
(y' )
whenever
x,x'
E
K,
y,y'
E
Wand
x+y = x'+y'). We have previously used this reduction in our analysis, as for example in proposition 2.56, where the semi-norms were used to obtain Cauchy estimates and the semi-norms were used to prove coherence.
If
H(O)
(*) (**)
is regular, then we
have Cauchy estimates and it is possible that this also implies coherence.
We are not,
however, able to prove this.
To prove coherence, we need extra hypotheses and these can take various forms.
One may place conditions on
as local connectedness, or conditions on ility or a combination of conditions on
E K
K,
such
such as metrizaband
E.
We shall
302
Chapter 6
assume that
K
is metrizable and that
E
satisfies a certain
technical condition which appears to be satisfied by most, not all,
spaces for which
H(O)
is regular.
if
This gives us
examples of non-metrizable locally convex spaces in which is regular for every compact set
H(K)
Our methods are
K.
easily seen to be influenced by the proofs of proposition 2.56 and theorem 6.1. Proposition 6.2
a)
K
b)
H(O)
c)
if
K
Let
E
convex space
be a compact subset of a locally
and suppose
is metrizable, is regular, is a convex balanced open subset of
V
for each n, (fn)nCH(V), f n =I- 0 there exists a bounded sequence in V,
then
and
f n(x ) f 0 n
such that then
(xn)~=l'
n
is a regular inductive limit.
H(K)
Proof
for aU
E
Let
B
be a bounded
subset of
H(K).
Since each
semi-norm of the form
P (f) where on
p
H(K)
is a continuous semi-norm on and
H(O)
a neighbourhood
for every
x
V
in
Now suppose nets in
K,
zero in
E,
such that
of
ex
is continuous
0
in
E
and
M> 0
such that
K, B
(x ) a aEr (Ya)aEr y ,y' E V ex
H(O),
is regular it follows that there exists
is not coherent. and and all
Then there exists two
two nets converging to (x' ) a aEr' and (f ) a net in B (y'a ) aE r' a aEr ex x' +y' x +y for all and ex, a ex ex ex
Germs, surjective limits,
€
303
-products and power series spaces
I)
Since
K
is metrizable
K-K
a
f 0
is also metrizable and hence the
{Ya-Y~}aEr contains a null sequence (Yn-Y~)~=l' Let (xn)~=l and (x~)~=l be the corresponding sequences in K.
set
For each positive integer n and each x in V 1\' 1\' dJf dJf (x') an n a (x n ) n h n(x) (x) - 2:~=0 (x+Y~-Yn) 2:;=0 , J' . J.
,
let
,
h n is a holomorphic function on V and hn(Yn) f O. By condition (c), there exists a bounded sequence (zn)~=l
Each
V
such that
hn(zn) f 0
for all
n.
in
By the identity theorem
for holomorphic functions of one complex variable we can choose a null sequence in 2Sn> 0
for all
such that
~,
n.
Hence
"nzn
-*
loss of,generality, we may suppose An Z n + Y~ - Yn E V
for a 11
n.
0
as
n
-*
00
"nzn E V
and, without and
Now c h 0 0 s e i n d u c t i vel Y a s t ric t 1 Y
increasing sequence of positive integers, kn 2 Sn > n a I l nand
such that
j !
j !
Let q(f)
for every If
f
f E H(K)
in
H(K).
then there exist a neighbourhood
and a po sit i v e in t e g e r n 0 Ilf II K+ 4 W ~ M,
An Z nEW
and
s u c h t hat
oo
f EH
An Z n + Y~ - YnEW
(
W of
0,
K+ 4 W) , all
n ~ no .
M>O
304
Chapter 6
Hence
(A n z n +y'-y)1 n n
:i
Since
is barrelled
H(K)
and
q (f
and B
n for all n an n this leads to a contradiction. Hence Since
k
is a continuous semi-norm.
q
kn
1 )
M .
This completes the
proof. If
is a fully nuclear space with a basis,
E
E
compact subset of
is metrizable.
osition 6.2 is satisfied by norm.
if
ES
then every of prop-
admits a continuous
Hence we have the following corollary to proposition
6.2 and this applies,
DN
E
Condition (c)
in particular,
to strong duals of nuclear
spaces.
Corollary 6.3
such that then
H(K)
subset
K
If
E
of
E.
is a nuclear locally convex space,
compact subset of then,
is a fuL Ly nucLear space with a basis
E
If
is reguLar admits a continuous norm and H(OE) E' S is a regular inductive limit for every compact
E
and
V
K
is a
is a neighbourhood of zero in
E
using Cauchy estimates, one can show there exists a
neighbourhood
W
of zero such that
H(K)
induce the same uniform structure and hence the same topology on the unit ball of space,
Hoo(K+W).
Since
HOO(K+W)
is a Banach
corollary 6.3 yields the following result.
Corollary 6.4
If
E
is a fully nucLear space with a basis,
Germs, surjective limits,
€
305
-products and power series spaces
E' admits a continuous norm and H(OE) is regular, then S H(K) is quasicomplete for any compact subset K of E. In chapter 2,
we defined
(definition 2.59)
the
T TI
ology on space. with
that
H(U)
for
U
top-
an open subset of a locally convex
This topology has good local properties and coincides T
indeed it has been conjectured
in certain cases, W
and
T
always coincide.
T
TI
We now examine this top-
W
ology and begin by showing that it is indeed well defined. Lemma 6.5
space
E.
lim
H(K)
u.
is the set of all compact subsets of
R(U)
Proof
be an open subset of the locally convex
Then algebraically
H(U)
where
U
Let
Under the natural restriction mappings is clearly a projective system.
{H(K) }KE 1«U) mapping
H(U)-----+ lim
A
The canonical
H (K) ,
+-
Kd::(U) where A (f)
=
([f]K) KE :k(U)
and
is the holomorphic germ on
[f]K
K
induced by
f,
is
linear and injective.
It remains to show that
A
ive.
E lim
We define a
Let
H(K)
be given.
is surject-
KEj«(U)
function claim
f
f
on
E H(U)
compact subset of
U
by
and U,
f(x) A(f)
=
=
f{x}(x)
(fK)KE:k(U)'
then since
E lim K E~(U)
H (K) ,
for all If
K
x
in
is any
U.
We
306
Chapter 6
r..n
d f {x} (x)
nl
nl
for any compact subset negative integer Hence if
x
E
U
of zero such that in
V
K
of
U
containing
x
and any non-
n. and
V
x+VCU
is a convex balanced neighbourhood and
f{x}
Hoo(x+V)
E
then for any
y
we have f(x+y)
f{ x+y } (x+y)
f [x, x+y] (x+y)
(\
dnf 00
In=o
[x,x+y]
( ) x (y)
nl
where
{x+>.y; O::A::l}.
[x,x+y]
This shows that set of
U,
X E
f K
E
H(U).
Moreover,
and
n
is arbi trary,
nl
K
if
nl
and consequently
f
K
is a compact sub-
then
nl Hence
.
This
A(f)
completes the proof. Remark 6.6
We have
TO'TIT'TW
on
H(U),
open subset of a locally convex space. describes a situation in which Proposition 6.7
If
locally convex space, Proof Suppose
Let p
p
be a
U then
TW
T
and
w
U
an arbitrary
Our next proposition T
IT
coincide.
is a balanced open subset of a T
T W
IT
on
H(U).
continuous semi-norm on
is ported by the compact balanced subset
H(U). K
of
U.
Germs, suriective limits,
€
-products and power series spaces
307
By theorem 3.22, we may suppose, without loss of generality, that 1\
dnf (0)
pC L~=o
,,'" 1,n=O pC
nl
L~=o
anfeO)
for every
nl
dnfCO)
O we such that for every E
of
V
with
WCB+aV.
Every normed linear space is quasi-normable and a locally convex space is a Schwartz space if and only if it is quasinormable and its bounded sets are precompact.
Thus a Frechet-
Montel space is quasi-normable if and only if it is a Fr~chet Schwartz space,
If
Proposition 6.18
is a compact subset of a quasi-
K
normable metrizable space
E
then
H(K)
-
lim (HOO(V), II
Ilv)
V:JK, V open
is a boundedly retractive inductive limit. Proof
We apply proposition 6.16.
sequence in
H(K).
Since
H(K)
Let
(fn)~=l
be a null
is a regular inductive limit
(proposition 2.55), there exists a convex balanced neighbourhood
V
of zero such that
sup Ilf II K+V = M
n
n
O
W
of zero,
2WCV,
we can find a bounded subset
B
such of
E
314
Chapter 6
with is a
WCB+aV. null
(fn)~=l
We complete the proof by showing that oo
sequence in
II
(H (K+W),
II K+W)
.
cimf (x) Since
I:=o
fn(x+y)
(y)
n
for every
x
in
K
m! and
y
in
W
and
dmfn
II for all
x
sup XEK Given
in
K
m!
and all
r:~('l
r,) n
----+0
choose
B
II
::c
M
2
W
m
it suffices to show
n-+ oo
as
bounded in
for each
E
m.
such that
WCB+ oV.
+
where
is the symmetric
with
Since m!
that
+
n
linear form associated I
YI+oY2-oY2EW+oVCZV +oVCV
we see
Germs. surjective limits.
Since
€
f
sup
p(f)
315
-products and power series spaces
E
H(K),
is a continuous
XEK
semi-norm on
for all
m
H(K)
and
and
n
this implies
Hence
as
n~oo
f
n
~
0
in
Hoo(K+W)
and this completes the proof.
Corollary 6.19
If
is an open subset of a quasi-normable
U
E
metrizable locally convex space
then
(H(U),T
) W
is
complete. By proposition 6.18,
Proof
for any compact subset
K
of
H(K) U.
is boundedly retractive
By proposition 6.15,
lim U:)V:)K V open is also boundedly retractive and hence complete. (H(U),. ) w
lim
(l im
+-
-7
Since
(proposition 6.12)
KCU U:::>v.=>K K compact and a projectivci limit of complete spaces is complete, shows that
(H(U),T ) w
is complete.
this
This completes the proof.
A weak converse to proposition 6.18 is also true as one can easily prove the following:
if
E
is a distinguished Frechet space and
H(K)
is boundedly V,:)K V open
retractive for some non-empty compact subset
Chapter 6
316 K
of
E
then
E
In particular, E
is quasi-normable.
H(OE)
is not boundedly retractive when
is a Fr~chet Montel space which is not a Frechet Schwartz
space.
SURJECTIVE LIMITS OF LOCALLY CONVEX SPACES
§6 •2
We now describe a method of decomposing spaces of holomorphic functions
into a union of more adaptable subspaces.
Alternatively,
this method may be described as a way of gener-
ating locally convex spaces with useful holomorphic properties. Our method, theorem,
the use of surjective limits and Liouville's
is based on the factorization results of chapter two
and arises naturally in many problems of infinite dimensional holomorphy. on
Its range of usefulness for problems of topologies
is not as great as in some other areas as for
H(U)
instance in solving the Levi problem.
A collection of locally convex spaces and
Definition 6.20
linear mappings
(E.,IT.). 1
1
lS
A
is called a surjective represen-
tation of the locally convex space tinuous Linear mapping from
E
E
onto
i f each E·
1
and
IT. is a con1_1 (IT. (V.)). A 1
1
lS
forms a base (and not a subbase) for the filter of neighbourhoods of
o
in
in
0
Ei
Limit of
as
E
and
(E.,IT.). 1
1
Vi
ranges over the neighbourhoods of
ranges over
i lS
A
A.
E
and we write
E
is caLLed the surjective lim (E;,IT ~
+-
·). 1
isA If each
ITi
is an open mapping, we call
~
(Ei,IT i )
isA an open surjective limit and if for each subset
K
such that
of
Ei
TIi(K ) i
isA
and each compact
there exists a compact subset =
K
then we say
lim (Ei,TI ) i
Ki
of
E
is a compact
+-
isA
surjective limit. Every locally convex space is a surjective limit of normed
Germs, surjective limits,
€
317
-products and power series spaces
linear spaces, nuclear spaces are surjective limits of separable inner product spaces and a locally convex space which has the weak topology is a surjective limit of finite dimensional spaces. Example 6.21 where
Al
TIiEA Ei
is a surjective limit of
ranges over all the finite subsets of
TI.
lE
A E.
1 1
This
A.
surjective limit is easily seen to be open and compact. Example 6.22 and
.£
X
If
is a completely regular Hausdorff space
is the space of all continuous complex valued
(X)
X
functions on
endowed with the topology of uniform conver-
gence on the compact subsets of
X,
then
.{be X) KCX
where
ranges over the compact subs ets of
K
fb (K)
and
X
is the Banach space of complex valued continuous functions on
K
endowed with the sup norm topology.
X
Since
is a complet-
ely regular space, the Tietze extension theorem implies that lim
is a compact surjective limit and the open
+-
KCX
mapping theorem for Banach spaces implies that it is an open surjective limit. Example 6.23
The strong dual of a strict inductive limit
of Frfchet Mantel spaces is an open and compact surjective limit of t>:Jm ·spaces. Proof
Let
E
lim (En"n)
be a strict inductive limit
---+
n
of Fr'chet-Montel spaces.
Since
E
induces on
En
its
original topology, we see, by the Hahn-Banach theorem,
that
the transpose of the canonical injection of
E
surjective mapping from on
E'
ES
onto
(En)S'
En
into
is a
The strong topology
is the topology of uniform convergence on the bounded
subsets of
E
and,
since each bounded subset of
E
318
Chapter 6
is contained and compact in some
En'
the topology on
ES
is
the weakest topology for which all the transpose mappings are continuous.
Hence
ES
((En)6)~=1.
is a surjective limit of
An application of the open mapping theorem shows that it is an open surjective limit. (E ) n
B
zero in E
En
whose polar in
Kn
(En)S'
is a strict inductive limit,
W of
E
in
0
be a compact subset of
and,
WO
Montel space
contains
Kn
of Since
there exists a neighbourhood
¢EVO then Hence if V::> wn E n by the Hahn-Banach theorem, there exists and
I'¢(W) I ::, I
such that
rEE'
Va,
V
such that
I
::,
1¢(WnEn)1 a
Now let
There exists a convex balanced neighbourhood
I.
't IE n
= ¢. of
is a compact subset
completes the proof.
a: N x a: (N)
In particular, we note that
E
As E I. S
is a
This
lim (; x B is an n-l O
polynomial which vanishes on LX ,,0.
Now let
defined by
y EES
LYCz)
induction, on E'
By our induction hypothesis Then
LY
B
ES'
VO
LY:L
S ->
be
[
is a hypocontinVO
which vanishes on
E'
In particular, we have
B'
y
•
be arbitrary and let
L(z,y, ... ,y).
uous linear form on for any
V
defined by homogeneous hypo continuous
,
and hence-by
=
LY (y)
p (y)
= 0
in analytic functions on
Hence
E' B
is an open and compact surjective limit,
lim CCEn)B,I1n)
U.
is a determining set for hypo-
Now let
f
E
Since
HHY(U),
+--
n
it suffices to show that
f
factors through some
were not true,
then for each integer
zn
E
and
zn+Yn
U
n
ID!Sn,
this
there exist
such that
o For each
n,
(E~)B.If
and
the function
z
->
f(z+Yn)-f(z)
defines a non-zero
hypoanalytic function on some convex balanced neighbourhood of
Germs, surjective limits,
zero in
ES
321
-products and power series spaces such that
and hence there exists
F f(x n )·
f(xn+Yn)
For all
n::: m
F gn(l),
the function
gn
which maps
AE
and hence we can choose
Hence
A n
E
a:
to
(:
is a non-constant entire function,
f(xn+Ayn)-f(x n ) gn(O)
€
since
such that
Since (Yn)~=m is a 1 n xnEV°(\2U for all
n::: m.
very strongly convergent sequence and it follows that of
{x +A y}OO is a relatively compact subset n n n n=m This contradicts the fact that f is unbounded on
U.
{x +A Y }OO and hence n n n n=m f EH(U). Now suppose H(U).
is a
ES
,
(E
example 2.47).
n
S
(En)S
is a
this would complete the proof (see
If this were not so,
the first part of the proof, relatively compact in
)' and
bounded subset of
is a compact surjective limit and
space for each
n
o factors uniformly through some
We claim that
(En)~Since ~jrQ
factors through some
f
U
then we could find,
as in
a sequence
and a sequence of functions
I fn ( x + a y ) I > n for a 11 n. Cfn)~=l CCfa)aEf such that n n n This contradicts the fact that (fa)aEf is '0 bounded and completes the proof.
If
Corollary 6.26 H (U) ,
,
p (f £
n
for all
n factors through some
B
,
0
bounded subsets of
E
n H (V) ,
,
and is also V
0
n.
Hence
bounded.
Since
iJ J- fYl
an open subset of a
(example 2.47),
space, are locally bounded
323
-products and power series spaces
this completes the
proof. It is worth noting that even though propositions 6.25 and 6.27 are similar in statement
(both hypothesis and
conclu~
ion), quite different methods are used to get uniform factorizations. Combining propositions 6.25 and 6.27 we obtain the following result.
E
Let
Proposition 6.28
lim E ---+
n
be a strict inductive
n
limit of Frechet Montel nuclear) spaces.
(resp.
Fr:chet Schwartz,
Fr~chet
Then (H(E~)" IJ
W,
b)
= lim(H(E )B')" n
~
0
)
n
is a strict inductive limit of Frechet-Montel Schwartz,
Frechet nuclear) spaces and the
of
are locally bounded.
H(E)
complete and i f
E
Moreover,
Frfchet space. then
If
En
(H(En)P"o)
corollary 3.38
is Mantel
is Montel
(resp.
Crespo
Fr~chet
(resp.
bounded subsets
w (H(ES)"o)
admits a continuous norm,
In example 2.47, we showed that
Proof
,
then
is
'0 = ,
o,b
is a Schwartz, nuclear),
Schwartz, nuclear), by
(resp. a modification of the proof of proposit-
ion 6.9, corollary 3.65). Since
E'=lim(E)' B n B +-
is an open and compact surjective
n
limit,
it follows that
lim ---+
n
is a strict inductive
324
Chapter 6
limit of Fr'chet spaces and consequently it is complete.
To
and the inductive limit topology on
H(E
) S
As
are both
bornological and have the same bounded sets, by proposition 6.27, they are equal. Our final
This completes the proof.
application of surjective limits is to holomor-
phic germs and in this example,
we do not assume that the
indexing set is countable. Proposition 6.29
fo
space and suppose space.
Then
X
Let
is an infrabarrelled locally conVex
is a regular inductive limit of Banach
H(K)
K
spaces whenever
(X)
be a completely regular Hausdorff
is a compact metrizable subset of
Proof
denote the set of all bounded contin-
uous functions on '£'b (X)
X.
By the Tietze extension theorem
is a dense subspace
of.e (X).
balanced open subset of .£,(X) xn s Vn ~b (X)
complex variable, that
If
V
is a convex
(fn)~=l C
and
H(g (X))
then
fn (x n ) f- 0 for all By the identity theorem for holomorphic functions of one
we can choose n.
ib(X).
I An I· II xn Ilx
Anxn ->- 0
as
such that
choose a sequence of scalars
~ ~
n->-oo
and
in,e, (X)
fn (>'nxn) and
osition 6.2 it suffices to show
f-
for all
0
f- O.
fn 0nxn) H(O)
(An)~=l
n.
such
Hence
By prop-
is regular to complete
the proof. Let
B
be a bounded subset of
n=O,I,2, ... ,
H(O)
and let
fE B}.
n!
It obviously suffices to show there exists a neighbourhood of zero in
£
(X)
such that
let Wp
{XEX; f,g
for all E ~(X)
P(f+g)
Vx
supllFllv
FEtr
open in
support(g)C:V x
f- P(f)}.
n
for all
n.
be the degree of the homogeneous polynomial
If
sup k n
N
n.
J
for
W is a compact
Hence
X.
We now claim that each F
I'
Since
we again get a contradiction.
subset of
If
J
B
then
L
of
F X
F
'"B
in
since
lim
+--
r9 (K)
l(W).
factors through
£
factors through
for ·some compact
(L)
is an open surjective limit.
KCX Let
f,g
hood of
£(X)
E
W.
and suppose
Choose
hI
E
If
x
E
such that
is equal to I hI (hl)e {x;g(x) = O}
such that
W
and support
VI then there exists a neighbourhood
L"-. VI
support
£, (X) of
on a neighbourhood
vanishes on some neighbour-
g
F(fx+g ) x C Vx·
(gx)
and support
= F(fx)
for any
Choose
(hx)CV x ·
hx
Let
E
fx,gx
x
jb(X),
E
such th~t
£,(X)
of
Vx
hx(x)
2"}.
Vx = {ysX;hx(y»
= I
The set
U contains
and hence there exists
WuL
'"xlv V WuLCV1UV x
such that such that
k(x)
= 1
all
x
xl,···,x n
tv
2
t-
UVx N
VI U
"" V
n
Xl identically zero on some neighbourhood of
k 1 (x) and
n
= k(x)+h 1 (x) +Li=lhx.(x) Ikl(x)1 >- ~ for ever} x Let
kj
k
i = I, ... , n. .-v
and
/\
L~=l h.l.
-
I
on
X
rv
UV
k
and
x
k
n K u L.
Let
XE X.
kI
E
X. -.J
hljl . J\l
hI
Now
k + hI +
and
1\
kl
for every in
L .......... V I
E
Now choose
hi
hx · l.jk
for I
g
Eg eX) is
(X)
327
Germs, surjective limits, €-products and power series spaces
F (f+g)
""k
(since
is identically zero on a K u L)
neighbourhood of
1\
(since
support
=
C{x,g(x)
(hI)
support
(hI)
O})
F(f) (since
support
'" g) C (hi
VX.
i=l, ... ,n).
Now suppose
g
~(X)
W
choose
1
on some neighbourhood of
of
V.
hv
vanishes on
Since
converges to
E:
gi K.
For each neighbourhood
W.
such that
hV
Wand
K = 0 the net Hence F(f+g)
hv
hv g =
=0
converges to zero as
through
W.
Sin c e l i mg (X )
Thus each
of
is identically
lim F(f+hV g )
K.
V
on the complement F (f)
V-+K
vanishes on a neighbourhood of
for
1.
F
N
E:
B
V
as factors
is a compact surjective limit and
--
f (x) .
o (x) (f)
0 s HHY(U; (HHY(U) "0) ~) Proof
f*
f*
be an open subset of a locally convex
U
and le t
E
=
(H(E~)"o)'
into
Lemma 6.32 space
and
v s(E~)'CH(E~),
Since
denote the evaluatim
Then
.
The I ocall y convex space ((HHY (U)" o)~) ,
(HHY(U) "0) Since
HHY (U) .
is complete
/\
for fixed function and
0
x,y A
->-
and
dnf(x)
An
L~=o
o (X+AY) (f)
f
and all
o(x+Ay)(f)
~
sufficiently small,
the
is holomorphic at the origin in
[
is a G-holomorphic mapping.
Let
K
be a compact subset of
o(K)C{fsHHy(U); IIfIIK (l}o equicontinuous subset of on
(y)
nl
o(K)
topology.
Clearly
ogyon
and the weak topology on
K
0IK
Since
it follows that
(HHY(U)"o)'
(HHY(U)"o)~
induced by
U.
o(K)
is an
and hence the topology
is equal to the weak
is continuous for the initial topol(HHY(U)"o)'.
Thus
0IK
is continuous and this completes the proof. Proposition 6.33 conVex space
and let
E
convex space.
given by
Let
o*(ljJ)
U F
be an open subset of a locally be a quasi-complete locally
The mapping
=
ljJoo
is a canonical isomorphism of locally convex spaces and hence
Chapter 6
330
By lemma 6.32,
Proof
8*
is well defined and it is
obviously linear and injective.
We now show that
surjective.
We define
Let
by the formula
f E HHy(U;F).
v(f*(w))
w(vof)
for every
6*
is
vEF'
and w in
If K is a compact subset of U then the (HHY(U)"o)'· closed convex hull of f(K),L, is a compact subset of F. v in F' and for fixed for every Hence II vof II K ( IIv II L the mapping
v E F'
-+
is continuous when
w (vof)
endowed with the compact open topology.
Thus
and
then
f*
is well defined.
v
If
6 E cs(F)
F'
w
is
f* (w) E
(F~)'
F
(vof)vEF',lvl(6
is a relatively compact subset of and f* is a continuous linear mapping. 6(f*(w)) '" Ilwllv Moreover, for any x in U and v in F' v(8*f*(x)) and so
6*f*
v(f*(6(x)))
=
f.
6 (x) (vof)
vof(x)
It remains to show that
6*
is a topol-
ogical isomorphism. Let all in 6
0:
K
be a compact subset of and let
V
U, let o:(h) " Ilh 11K be the polar of the 0: unit ball
hE HHY(U), If 6ECS(F) (HHY(U) " 0 ) ' · unit ball in F' , then sup{ I v (f* (w)) I ~ WEV ,v EW}
1:.6 (f*)
and
W
is the polar of the
sup{ Iw(vof) l,wEV,vEW}
sup 6(f(x)) XEK for any
f
and
f*
as defined above.
This completes the proof.
331
Germs, surjective limits, €-products and power series spaces Proposition 6.34
Let
locally convex spaces HHY(UxV)
=
U
and
and
E
=
HHY(U) ~ HHY(V)
topologically
V
be open subsets of the
respectively.
F
HHy(U;HHY(V))
Then
algebraically and
(each function space is given the compact open
topology). Proof
Since
(HHy(V),T
6.33 implies that f rJ
=
f(x) (y)
is quasi-complete, proposition
)
£ HHY(V)
HHY(U)
HHy(U;HHY(V)),
=
-oJ
We define
HHY (U x V).
E:
O
f:U + HHY(V)
Now let
by the formula
By using the Cauchy integral formula one
f(x,y).
sees that the mapping
(x) nl
is hypoanalytic for any fixed negative integer
n.
Xo
in
U,
and any non-
Hence the function
An d f(xo'Y) xEE ->- [y +
(x)
]
nl
belongs to
Since lin
"-'
L~=O
f (xo + AX) (y)
An n d f(xo'Y) Ln=O A ----~---(x)
,y)
d f(x
,00
(Ax)
0
nl
nl
for any fixed
Xo
in
U,
x
in
E,
sufficiently small it follows that Now let
K
respectively. continuous.
0.
for all
L x
0.
E:
for a given
- f(x,y)!
f
y E:
E
V
and
~
and all
U
and
V
is continuous it is uniformly K
+
n>O
x
as
0.+ 00
we have
n
y
A
HG(U;HHY(V)),
be compact subsets of
f!KxL
Hence if
such that !f(x ,y)
and
Since
,..,
in
L.
Thus
then there exists
332
Chapter 6
'" f(x)II
Ilf(x ) a
rV
= and
l'
10
-
suplf(x ,y) YE:L a
[(x) (y) I
f(x,y) I
~
n
all a
~
a
'it
on
UxV
V
respect-
o
HHy(U;HHY(V)) .
Now suppose the formula
g
HHy(U;HHY(V)),
10
g(u,v)
K and
Let ively.
-
sup I f (x ) (y) YE:L a
L
Let
ua
L
10
We define
by
g(u)(v).
be compact subsets of
K + u
and
v 13
10
U
a,13+ oo
as
L+ v
and
respectively.
Then
g(u) (v)
~
II g (u a)
-+
g (u) II L
as
0
since
-
a, 13
g(u)1 L
->-
Ig (u) (v 13)
+
-
g (u) (v) I
00
g(u a ) +
is continuous and
uniformly on the compact subsets of
Hence HHY(UXV)
gE: HHY(UXV) with
and we may algebraically identify
HHy(U;HHY(V)),
this is also a topological
Let
conVex spaces and let space.
If
U,V
as a+ oo
V.
sup li(u, v) I ue:K,ve:L
Corollary 6.35
g(u)
and
U F
uxv
sup II g (u) IlL ue:K
isomorphism and completes the proof.
and
V
be open subsets of locally
be a quasi-complete locally convex
are k-spaces, then
333
Germs, surjective limits, € -products and power series spaces
and H (Ux V) ,
(H(U;(H(V),T )LT )
TO)
o
Corollary 6.35 applies if
U
and
V
0
are both open subsets of
Frechet spaces or both are open subsets of JY1·m
spaces.
In
/'.
our next proof, we use the fact that
E f: F = E
0£ F
if
E
is
a locally convex space with the approximation property (see Appendix II). As our first application of the
E.
product, we prove a
converse to theorem 5.42.
E
Theorem 6.36
and
TI)
Proof
Let
P
HeE)
on
is a Frechet nuclear spaoe with a basis then
(en)~=l
and
and
E,a:
DN
space.
E
be an absolute basis for
E
be the closed subspace of
E = 0: x F
is a
E
spanned by
and let
(en)~=2'
Since
Fare Frechet nuclear spaces, an
application of Corollary 6.35 shows that
If
T8
Hee)
=
@-f..
TO
on of
pI
H(E) H(E)
then the closed complemented subspace is a bornological space.
By proposition
15, chapter 2 of A. Grothendieck's thesis, this implies that
P
contains an increasing fundamental system of weights
(w m):=1'
wm
= (wm,n)~=l'
supC\x n n
such that w
~
\w m,n )€:
is finite for every positive integer
(Xn)~=l
F.
Letting
\w m,n .(
)
in
p
=!
::i
sup \ x
m,
all
and taking
€>
pth
0
and all
roots we see
that sup\x n n
p W
m,n
n
n
\. W
1 .( m+, n
Wm+ 1 W
,n
m,n
)
p
334
Chapter 6
for all positive integers
m
and
p
and all
in
F.
Hence
)p}
00
n=l
w m,n is a continuous weight on F
is a
that
ON
E
space.
is also a
F
Since ON
and proposition 5.40 implies that
=
E
II: x Fe. sxs -;! s
this means
space and completes the proof.
Theorems 5.42 and 6.36 together give the following:
is a Fr~chet nuclear space with a basis then
E
If
on
TO = To
E
i f and only i f
H(E)
is a
ON
space.
For our next application, a kernels theorem for analytic
functionals on certain fully nuclear spaces, we need a further E~F
spaces we let E
6
F
E
If
type of tensor product.
and
are locally convex
F
denote the completion of the vector space
endowed with the topo logy of uniform convergence on the
separately equicontinuous subsets of the set of all separately continuous bi linear forms on If
Proposition 6.37
complete nuclear
ON
(H(UxV) ,T )' o
Proof
Since
nuclear
ON
S
·ot
U,V
spaces, and
U
Ex F.
and
V
are open polydiscs in
spaces with a basis, (H(U) ,T )' 0(H(V)
S
0
and
U
x
V
,T
0
)'
S
then M ?i H(UM)Q
is satisfied,
Hence
a
a q-l Z n
Zqn
1
implies
Cz
azq -In
(C ) q Z
::;
Zn
a
n.
for all
To complete the proof, we
is equivalent to an increasing sequence. show that it is an increasing sequence along the
arithmetic progression
and then modify certain inter-
mediate values to obtain an equivalent increasing sequence. Now a
kn
a
a
(kn) -p
_kn k -p ~
n-p
n
a
n
and hence the sequence a
sup n
kn
a
n
For
1
is increasing. n ::;
::; nk
we have
Let
Chapter 6
350
Ct
kn
n j
since
~
and
Por any positive integers
n
with
~
I.
kn ,
we
let
00
Since the sequence all
n
(Ykn)n=l
(Yj)j
we see that
Por any
nand
is increasing and
Y n k
I
is an increasing sequence.
j, k n ~j ~kn+l
we have
y.
and
J
Hence
Y
and
are equivalent
Y
sequences and this completes the proof. 6.52
If Ct
I
inf n
l
CnCk)+l) log eC
4
such that
'" nCk)
~
2n
we have
is the smallest power of
Since
)
for each positive integer
is an increasing function,
m
2n
q logCk+l)
for all
k.
35~
Germs, surjective limits, €-products and power series spaces
for all
k.
Thus for all
k
with
n(k)
n
we have
I
I
~
(a
l
(a I
nan
... ann!A*(n))n
...
a
n
n!
n
a
n!
n(k)an(k)
I )n
an n
n
;,
l
...
a
n
I q(log(k+l))a[
I
q
log(k+l)]
I
and
(Ct
I
.,.
Ct
n
n!
A*(n+l))n
1 n
(n+l)n+l
)
(n+l)!
(n + 1 ) an + 1 =
(n (k ) + 1 ) a n( k ) + 1
Zq (log(k+l))a[Zq log(k+l)]'
Since
a
is stable,
it follows
that there exists
C' > 0
such
that
1
C' (log(k+l))a[log(k+l)] :;
F
-1
(k):; C'(log(k+l))a[log(k+l)]'
This completes the proof.
Example 6.53
(a)
Let
a
(n P )
Since Zp
we have
00
n=l
where
p
is positive.
354
Chapter 6
sup n
a
n
By theorem 6.52,
°
where
[log(n+l)]
n
p+l
for all
n.
In particular,
/\J(log (n+l)) for any positive integer (b)
m.
a = (Pn)~=l
If
where
Pn
denotes the
nth
prime
then the fundamental theorem on the density of primes shows
on =(log log(n+l))(log(n+l)) (c)
If a
p>O
and
e
n
O-B.
IT
Show that
:E
-+
Eo
a,S a Show that a
~
directed surjective limit of Frechet spaces is an open surjective limit. 6.69
If
uous basis,
E
is a locally convex space with an equicontin-
show that
E
is a surjective limit of normed
359
Genns, surjective limits, €-products and power series spaces
linear spaces, also that spaces,
Show
each of which has an equicontinuous basis. is an open surjective limit of locally convex
E
each of which has an equicontinuous basis and admits a
continuous norm.
6.70*
By considering the space
show that in general
Co (r),
uncountable,
r
bounded subsets of
TO
H(E)
=
lim Ei' +
i
do not uniformly factor through some
E.1
even when we are
dea~
ing with an open and compact surjective limit.
6.71
Let
V
be a Reinhardt domain,
containing the origin,
in a Banach space with an unconditional basis.
6.72*
Let
K.
where each
be a Fr~chet-Schwartz space.
E
is a compact subset of
J
that there exists
T. J
n
T
E:
H(K.)' J
Let
If
E.
for each
Show that
T
K E:
=
U j -1
H (K) ,
K.
J
show
such that
= Lj=l T j .
6.73*
If
show that
K
is a compact subset of a locally convex
H(K)
spa~e
lim
-->-
V::::>K,V open
6.74
Show that the E -product of two
Jj1J
spaces is again
aJJ1~space. 6.75*
If
is a ~JJ
E
space and an inductive limit of
Banach spaces with the approximation property via compact mappings,
show that
every compact subset
6.76*
Let
H(K)
K
has the approximation property for of
E.
K
be a compact subset of [n is polynomially convex in for each
polynomials on
6.77*
Let
[N
i\(P)
a;N
n
E:
and suppose N.
are sequentially dense in
lin (K)
Show that the H (K) •
be a stable nuclear Frechet space which
admits a continuous norm.
Show that
Chapter 6
360
6.78*
Let
a
a
= nP(log(n+l))q
n
positive real numbers. where
6.79*
Show that
on = (log(n+l))P+
Let
1\ (P)
of weights P.
1
where
p
and
(H(l\oo(a)p,1) 0
(log log(n+l))q
for all
n.
be a sequence space with saturated system
Show that
is a Schwartz space if and only
f>.(P)
P there exist (a ~)~=l E P u)OO E C + such that a :;; u a' for all n. ( nn=l 0 n ~nn I\(P) is a Montel space if and only if for each
if for each
and each subsequence of integers
(an)~=l
E
I\(P)
there exists
an. inf __ J
1 im
such that
(nj)j=l
and Show that
O.
a~.FO,j""oo a~.
6.80*
Let H (U)
J J be a Fr~chet Montel space.
f>.(P)
for any open subset (H (~) ,
®
U
of
6.81*
Show that
6.82
Show that a locally convex space
topology
a(E,E')
TO) ,
(H (2)) ,
Show that
A(P)S'
TO)
S~ E
(H Cot) @~)
, TO)
S.
with the weak
is an open surjective limit of finite
dimensional spaces.
§6.6
NOTES AND REMARKS The completeness of
locally convex space
[503].
H (K) ,
E,
He showed that
K
was first H(K)
a compact subset of a investigated by J. Mujica
is complete whenever
metrizable locally convex space with property cises 6.62 and 6.63).
K.D.
(B),
Bierstedt and R. Meise
E
is a
(see exer-
[69] proved
the same result for compact subsets of a Frechet Schwartz space and subsequently P. Aviles and J.
Mujica
[41]
extended this
result to quasi-normable metrizable locally convex spaces. general result that
H(K)
is complete for any compact subset
of a metrizable locally convex space, theorem 6.1, S.
Dineen [200).
The
is due to
361
Germs, surjective limits, €-products and power series spaces
Proposition 6.2 is due to R. aries 6.3 and 6.4 are due to S.
Soraggi
[669] while coroll-
Dineen [200].
Further examples
including proposition 6.29, concerning the regularity of when
H(K)
is a compact subset of certain non-metrizable
K
locally convex spaces are given in R.
Soraggi
[667,668,669] ..
From the viewpoint of holomorphic germs and analytic functionals,
[501] is also of inter-
the following result of J. Mujica
est:
if
K
is a compact locally connected subset of the met-
rizable Schwartz space
E
lim En'
where each
En
is a
+--
n
normed linear space and the linking maps are precompact, then for each continuous linear functional exists a sequence (i) (ii)
Vm
f
(iii)
in
if
d) (mE) )
H (K)
there
of vector measures such that
(Vm):=l
,e (K;
E
on
T
,
for all
m;
,00
1 Lm=O m!
for every
H(K); as an element
is the norm of
of
-l,(K;9(m E ))' l/m n limllvmllm O.
then for each
n,
m-+oo
Conversely, satisfying
given a sequence defines an element of
(i) and (iii), then (ii)
H (K) , .
Proposition 6.7 is due to S.B.
6.8 was discovered by A.
Chae
Baernstein [42],
[120].
Proposition
in his work on the
representation of holomorphic functions by boundary integrals. Proposition 6.9, K-D.
theorem 6.10 and corollary 6.11 are due to.
Bierstedt and R. Meise
[70].
See also E.
for a further proof of proposition 6.9.
R.
Nelimarkka [525]
Meise has recently
shown that T = T on any open subset of a Frfchet nuclear o 11 space and thus the basis assumption in corollary 6.11 is not necessary.
Example 6.13 is due to M.
used it to prove corollary 6.40.
Schottenloher
[644] who
Corollary 6.40 is also due
Chapter 6
362
independently, and by a different method, L.
Nachbin
to J.A.
Barroso and
The proof given here is slightly different
[53].
from either of the above. The regularity and completeness of inductive limits is extensively discussed in the literature, recent survey of K. of K-D.
Floret
[238],
see for instance, the
and the first few sections
[70], and has led to the defin-
Bierstedt and R. Meise
ition of many special kinds of inductive limits. research
[503]
has led him to define "Cauchy regular" inductive
limits and this concept, R. Meise
[70],
J. Mujica's
as pointed out by K-D.
Bierstedt and
coincides with the concept of boundedly retract-
ive inductive limits in the case of an injective inductive limit of Banach spaces.
H.
Neus
showed that many of
[527],
these concepts coincide for countable inductive limits of Banach spaces,
and proved proposition 6.16.
is an abstract version, of a result of J.
[69],
inductive limit
lim
due to K-D. Mujica
Bierstedt and R. Meise
[503].
(Hoo(V) (\ H(U),
Proposition 6.15
II
The idea of using the
Ilv)
is due to J. Mujica
--+
KC.VCU
who proved proposition 6.12 and used it to prove propo-
[503]
sition 6.18 and corollary 6.19.
Surjective limits are due independently to S. 190]
and E. L igocka
system). Their basic properties, Further references are P. [207],
Ph.
[463,467]
Noverraz and R.
Berner
[552],
Soraggi
due to L.A. de Moraes
M.
S.
Dineen
Schottenloher
Examples 6.21,6.22,6.23 and
Dineen [190].
Proposition 6.25 is
while a particular case of this
independently, to P.J.
Proposition 6.27 is due to P. R.
[443).
Schottenloher [640], M.e. Matos
[669).
[498],
and
[58,59,60,61,62],
S. Dineen, Ph. Noverraz and M.
lemma 6.24 are given in S. result is due,
[189,
examples and applications to
infinite dimensional holomorphy are given in [190] [186,189,191,193],
Dineen
[443], (who uses the terminology basic
Berner
Soraggi proves proposition 6.29 in
Boland and S. [61]
Dineen[91].
and S. Dineen [194).
[669].
In studying vector valued distributions,
L.
Schwartz
[648]
363
Germs. surjective limits. €-products and power series spaces compensated for the absence of the approximation property by
€: - products (definition 6.30). M. Schottenloher [631] introduced E. -products as a tool
defining
in infinite dimensional holomorphy. In [639] he
proved lemma 6.32, propositions 6.33,6.34, coroll-
ary 6.35 and gave example 6.31. 6.34 is due to A.
Hirschowitz
r
A weak form of proposition 43 , proposition 3.4] and
weighted versions of the same proposition are given in K.
Bierstedt
Theorem 6.36 is new.
[66,p.44 and 55].
The idea
of using tensor products and the connection between this theorem and proposition 15, chapter 2, of A. Grothendieck was pointed out to the author by D. Vogt.
[287]
Earlier a direct
counterexample, which applied to the nuclear power series space case, was given by S.
Dineen [202],
(see exercise 5.82).
It
would be of interest to extend this counterexample to the general case (it is our belief that this is possible) give a completely self-contained proof.
basis hypothesis in theorem 6.31 is necessary. 6.37 is due to S. Dineen [202]. 6.39 are due to K-D. applications of
£
and thus
We do not know if the Proposition
Proposition 6.38 and corollary
Bierstedt and R.
Meise [69,70].
-products in infinite dimensional
Further h~lomorphy
and kernel theorems for analytic functionals may be found in K-D. B.
Bierstedt and R. Meise [69,70]
Perrot
and in J.F.
Colombeau and
[157,158,159,161,162].
All the results of section 6.4 are due to M.
Borgens, R.
Meise and D. Vogt and most of them are contained in comprehensive paper, partially summarised in [95],
[96]. This contains
many further interesting examples of structure theorems for H(AooCa)S)' [97]
The same authors have written a further article
on the A-nuclearity of spaces of holomorphic functions
using refinements of the techniques developed in [96]. The symmetric tensor algebra (definition 6.54) was introduced by A. Colojoara [139]. theorem 6.55 for
DF
She proved an abstract form of
nuclear spaces but did not establish a
connection between her results and holomorphic functions. was done in
[96]
and detailed in [487].
This
Chapter 6
364
The results and methods of section 6.4 are still in the process of finding their final
form and very recent develop-
ments suggest that they will playa very important role in the future of the subject. D. Vogt
We shall only mention that R. Meise and
[485,486] have recently obtained a holomorphic
criterion for distinguishing open polydiscs in certain nuclear power series spaces and have shown in [489] topologies
'o"w
and
'8
on
H(A(P)),
that the three
A(P)
a fully nuclear
spcace with a basis, can all be interpreted as normal topologies in the sense of G.
Kothe
[397].
Appendix I
FURTHER DEVELOPMENTS IN INFINITE DIMENSIONAL HOLOMORPHY
In this appendix, we provide a brief survey of some research currently being developed within infinite dimensional holomorphy.
The topics we dis-
cuss emphasise the algebraic, geometric and differential, rather than the topological aspects of the theory.
We hope this introduction will inspire
the reader to further readings and to an overall appreciation of the unity of the subject. THE LEVI PROBLEM We begin by looking at a set of conditions on a domain convex space
U in a locally
E.
(a)
U is a pseudo-convex domain;
(b)
U is holomorphically convex.
(c)
U is a domain of holomorphy.
(d)
U is the domain of existence of a holomorphic function;
(e)
The
Cf)
If
a ~
problem is solvable in
U;
is a coherent analytic sheaf, then
Hl(U;;1)
=
O.
All these conditions are equivalent when
E
is a finite dimensional
space (see L. Hormander [347] and R. Gunning and H. Rossi [294]) and this equivalence may be regarded as one of the highlights of several complex variable theory.
Note that condition (a) is metric, (b) geometric, (c) and (d)
analytic, (e) differential and (f) algebraic.
In the case of
E
en,
the
classical Cartan-Thullen theorem [118], published in 1932, asserts that (b) and (c) are equivalent. equivalent for domains in
In 1911, E.E. Levi [441] asked if (a) and (d) were (2.
This became known as the Levi problem and
was solved by K. Oka [558] in 1942 and extended to domains in 365
[n
by K. Oka
366
Appendix I
[559]
in 1953 and by F. Norguet [530] and H.Bremermann
The implication
(f)
=>
(e)
[101] in 1954.
is due to P. Dolbeault [208], (b)
due to H. Cartan [115] and (a)
=>
=>
(f) is
(e) is proved by L. Hormander [346].
Attempts to extend these conditions and to show their equivalence on arbitrary locally convex spaces have never been routine and have led to many interesting developments and results.
We now describe the evolution
of this line of research in infinite dimensions together with some related topics such as plurisubharmonic functions, envelopes of holomorphy, etc.
[103] in 1957 was the first to consider pseudo-convex
H.J.Bremermann
domains, domains of holomorphy and plurisubharmonic functions (see proposition 4.12) in infinite dimensions. space to be pseudo-convex if the boundary of
He defined a domain (dU(x)
U in a Banach
x to U is plurisubharmonic and showed that this was equival-
U)
-log d
ent to the finite dimensional sections of
is the distance from
U being pseudo-convex.
In 1960
he showed that the envelope of holomorphy of a tubular domain in a Banach space was equal to its convex hull [104] and afterwards [105] extended a number of his results to linear topological vector spaces. C.O. Kiselman [381] proved that the upper regularization of a locally bounded countable family of plurisubharmonic functions on a Frechet space was plurisubharmonic (this is known as the convergence theorem) and this was extended to arbitrary families on complete topological vector spaces by P. Lelong [425], G. Coeur6 [126,129] and Ph. Noverraz [536]. In [425] P. Lelong began a systematic study of the basic properties of plurisubharmonic functions and polar sets in topological vector spaces. This direction of research is developed in P. Lelong [426,428,429,430,434],
G. Coeure [127,128,129], H. Herve [326] and Ph. Noverraz [536,537,538,545]. By using multiplicative linear functionals, H. Alexander [5] showed, in his thesis, that a domain extension
,...;
U,
spread over
U in a Banach space E,
E
admits a holomorphic
which is maximal with respect to the prop-
erty that the canonical mapping of ological isomorphism. if and only if
U
E
(H(U) "0) into (H (U) " 0 ) is a topHe also noted that (H(U)"o) is a barrelled space
is finite dimensional and thus could not conclude that
was the natural envelope of holomorphy of
U.
J.M. Exbrayat [233] is
the only accessible reference for Alexander's unpublished thesis.
Further developments
367
The next contributions are due to G. Coeur~ [128,129].
He defined
pseudo-convex domains spread over Banach spaces by using the distance function and showed that a domain spread
X is pseudo-convex if and only
if the plurisubharmonic hull of each compact subset of
X is also compact.
lhis result was extended to locally convex spaces by Ph. Noverraz [544]. To overcome the inadequacies of the compact open topology encountered by Alexander, Coeur~ defined the
'8
topology on domains spread over separ-
able Banach spaces and showed that any holomorphic extension of a domain leads to a
'8
topological isomorphism of the corresponding space of holo-
morphic functions.
This result was later extended to domains spread over
G. Coeur~ also proved
arbitrary Banach spaces by A. Hirschowitz [338,343]. in [129] that a suitable subset
~(X)
of the
X a domain spread over a separable Banach space
the structure of a holomorphic manifold spread over a holomorphic extension of holomorphy and
H(X)
'8
E,
E
X and that, furthermore, if
separates the points of
spectrum of
H(X),
could be endowed with and identified with X is a domain of X '?! ~ (X) .
X then
In 1969, two important contributions were made by A. Hirschowitz [335, In [335], he showed that the Levi problem had a positive solution
336] .
for open subsets of Riemann domains over II: 1\,
1\
eN
and this result was subsequently extended to
[N
by M.C. Matos [456] and to domains spread over
arbitrary, by V. Aurich [33].
In his analysis, A. Hirschowitz
showed that any pseudo-convex open subset
U of
eN
had the form
U = n-l(n (U)) for some positive integer n where n is the natural n n n projection from [N onto en. This result, together with factorization properties of holomorphic functions on
II:N
given by A. Hirschowitz in [335]
and also obtained independently by C.E. Rickart [605], led eventually to the concept of surjective limit (see §6.l) and to a technique for overcoming the lack of a continuous norm in certain delicate situations.
V. Aurich
used the bornological topology associated with the compact open topology in his investigation of the spectrum of
H(U),
U a domain spread over
[1\
[33] . In [336], see also [337] for details, A. Hirschowitz showed that the unit ball of
kS ([0, rI]) ,
rI
the first uncountable ordinal, is not the domain
of existance of a holomorphic function, i.e. (c)
r>
(d).
This counter-
example to the Levi problem and B. Josefson's [358] example of a domain in cocr),
r
uncountable, which is holomorphically convex but not a domain of
Appendix I
368
holomorphy, i.e. (b) "I> (c), rely heavily on the non-separability of,&[O,r2] and
r
respectively and, indeed, it appears that countability assumptions
have always, and probably always will, enter into solutions of the Levi problem.
We note in passing that A. Hirschowitz introduces bounding sets
in [336] and that this concept had also arisen in H. Alexander's work on normal extensions
,
in S. Dineen's investigation of locally convex top-
ologies on spaces of holomorphic functions, [177,178], and in M. Schottenloher's study [631] of holomorphic convexity. In three further papers [338,340,343], A. Hirschowitz looked at various other aspects of analytic continuation over Banach spaces.
He
showed, using germs of holomorphic functions, that every domain spread over a Banach space has an envelope of holomorphy.
His investigation of vector-
valued holomorphic functions showed that whenever
(
valued holomorphic
functions can be extended then so also can Banach valued holomorphic mappings and that conditions (b) and (c), (resp Cd)), remain unchanged when holomorphic functions are replaced by Banach (resp. separable Banach) space valued mappings. In 1970, S. Dineen [176] replaced morphic functions on E contained in Since
Hb(U)
H(U)
by
Hb(U),
the set of holo-
U which are bounded on the bounded open subsets of
U and at a positive distance from the boundary of
U.
has a natural Fr~chet space structure he was able, by suit-
ably modifying conditions (b) and (c), to obtain a Banach space version of the Cartan-Thullen theorem.
This approach was developed by M.C. Matos
[457,459,460] who proved similar Cartan-Thullen theorems for various subalgebras of
H(U).
Independently of S. Dineen [176] and A. Hirschowitz [338], M. Schottenloher [631,632] was considering a much more general situation by defining regular classes (see also G. Coeure [129]) and admissible coverings for domains spread over a Banach space. Cartan-Thullen theorem.
For each regular class he proved a
By looking at all regular classes and by general-
izing the classical intersection theorem for Riemann domains to infinite dimensions, he showed that the envelope of holomorphy could be identified with a connected component of the
'0
spectrum.
In [640], he extended
this result to domains spread over a collection of locally convex spaces which included all metrizable spaces and alldBfnq spaces, (see also K. Rusek and J. Siciak [618]).
In later papers, [633,635,638,639] he
369
Further developments
considered various other topics on analytic continuation in infinite dimensions such as vector-valued extensions and the extension problem for Mackey holomorphic functions. We now digress a little to describe more recent results on the spectrum of
H(U).
In [352], M. Isidro showed that
Spec(H(U),T ) ~ U when o
U is a convex balanced open subset of a complete locally convex space with the approximation property and this result was extended to polynomially convex domains in quasi-complete spaces with the approximation property by In [504], J. Mujica proved that Spec(H(U),T 6) ~U U is a polynomially convex domain in a Frechet space with the
J. Mujica, [502,505]. when
bounded approximation property and the counterexample of B. Josefson [358] shows that this result is not true for every po1ynomia1ly convex domain in Banach spaces with the approximation property. M. Schotten1oher proved Spec(H(U),T o ) = U for U pseudo-convex in a Fr{chet space with basis. The study of the spectrum is the study of the closed maximal ideals and a few authors have also studied finitely generated ideals in J. Mujica shows in [505] that
H(U),
U a po1ynomia11y convex domain in a
Fr~chet space with the approximation property, is the
ideal generated by any finite family of functions in zero.
H(U).
Tw
closure of the
H(U)
without common
In [277], B. Gramsch and W. Kabal10 prove the following result:
A is a Banach algebra with identity domain in a JJJ-tS
e,
if
U is a po1ynomial1y convex
space with Schauder basis and
(f j )j=l CH(U)
have the
property that for every x in U there exists (aj x)j=l C A such that n n ' Lj=l aj,Xfj(x) = e then there exists (aj)j=IC:A such that
L~=l ajfj(x) = e for every x in U. In particular, this shows that the ideal generated by any finite family of holomorphic functions without common zero in
U· is equal to
H(lJ)
(see also M. Schotten1oher [646]).
Further results and examples on analytic continuation, the spectrum of H(X),
Cartan-Thul1en theorems and the envelope of ho1omorphy are given in
the book of G. Coeurf [131]. We now return to our main theme.
The following fundamental property
of pseudo-convex domains in a locally convex space is due to S. Dineen, [183,186] and Ph. Noverraz [540, 544];
if
U is a pseudo-convex (resp.
finitely po1ynomial1y convex) open subset of a locally convex space p
E:
cs(E),
!l
is the natural surjection from
E onto
E/ker(p)'
and
E, !leU)
370
Appendix I
has non-empty interior then sections of
II(U)
U = II
-1
(II (U) )
and the finite dimensional
are pseudo-convex (resp. polynomially convex).
Various other forms and refinements of the above are known and they allow one to transfer problems, such as the Levi problem, from
U to
II(U)
and to generate locally convex spaces with preassigned properties. In [175], S. Dineen replaced the H(U)
and on showing
tw
to
topology by the
tw
topology on
to (theorem 4.38) obtained a Cartan-Thullen
theorem, i.e. (b)(c), for balanced open subsets of a Banach space with an unconditional basis.
The following year, S. Dineen and A. Hirschowitz
[203] improved this result by showing that a domain
U in a Banach space
with a Schauder basis is a domain of holomorphy if its finite dimensional sections are polynomially convex.
This result was extended to separable
Banach spaces with the projective approximation property by Ph. Noverraz [540,543,546] to metrizable and hereditary Lindelof spaces with an equiSchauder basis by S. Dineen [1861, to
JJJ~
spaces with a basis by N. Popa
[586] and to various other spaces by R. Pomes [583,584].
S. Dineen also
showed in [186] that the collection of spaces for which this result was valid was closed under the operation of open surjective limit. In [179], S. Dineen showed that an open subset of a Banach space with a Schauder basis is polynomially convex if and only if its finite dimensional sections have the same property.
This result was extended to Banach
spaces with the strong approximation property by Ph. Noverraz [540,544] and to various other spaces, including nuclear spaces, by using surjective limits in S. Dineen [183,186] and Ph. Noverraz [540,544].
All these
results are contained in the very general result of M. Schottenloher [643} who proved that the same equivalence was valid in any locally convex space with the approximation property. We now look at two closely related questions concerning polynomials, Runge's theorem and the Oka-Weil theorem. polynomials are dense in subset of
(n,
(H(U),t o )'
if and only if
Runge's theorem states that the
U a holomorphically convex open
U is polynomially convex while the Oka-
Weil theorem states that a holomorphic germ on a polynomially convex compact subset nomials.
K of
[n
can be uniformly approximated on
K by poly-
371
Further developments
In [605], C.E. Rickart proved an Oka-Weil theorem for
[A.
S. Dineen
[179] extended Runge's theorem to Banach spaces with a Schauder basis and in collaboration with Ph. Noverraz [539,54lJ proved an Oka-Weil theorem for the same class of spaces.
C. Matyszczyk [469] showed that the polynomials
are sequentially dense in
(H(U;F),T O)
open subset of
E
and
E
approximation properly.
and
F
when
U is a polynomially convex
are Banach spaces with the bounded
The next set of contributions were made indepen-
dently by Ph. Noverraz [540,543,546], S. Dineen [183,186], R. Aron and M. Schottenloher [31] and E. Ligocka [443].
Noverraz proved Runge's theorem
and the Oka-Weil theorem for locally convex spaces with the strong approximation property, while R. Aron and M. Schottenloher [31] proved a vector valued Runge theorem for domains in Banach spaces with the approximation property.
Ligocka proved an Oka-Weil theorem for locally convex spaces
which could be represented as a projective limit of normed linear spaces with a Schauder basis and this result included those of Dineen.
E. Ligocka
also showed that any polynomially convex compact subset of a complete locally convex space had a fundamental neighbourhood system of polynomially convex open sets.
J. Mujica [502] pointed out that Ligocka's proof extends
to quasicomplete spaces and hence for this collection of spaces the OkaWei 1 and Runge theorems are equivalent (see also Y. Fujimoto [249]).
In
[470], C. Matyszczyk proved an Oka-Weil theorem for Fr~chet spaces with the approximation property and this was extended to holomorphically complete metrizable locally convex spaces by M. Schottenloher [643].
In [502],
J. Mujica obtained a very general result by proving the Oka-Weil theorem
for quasi-complete locally convex spaces with the approximation property and applied this result to characterise the polynomially convex.
spectrum of
H(U),
U
Further approximation theorems are given in C. Maty-
szczyk [470] and J. Mujica [504]. E. Ligocka [443] is still open;
The following subtle problem posed by if
subset of the locally convex space subset of
TO
1\
E (the completion of
K is a polynomially convex compact E
is
K a polynomially convex compact
E)?
The study of the Levi problem led during this period to the investigation of concepts such as holomorphic completion (see section 2.4), pseudo-completion, w
spaces, etc.
We refer to Ph. Noverraz [540,543,544,
546,547], M. Schottenloher [633,637,645], S. Dineen [184,186] and G. Coeur~ [135] for details.
These topics and fundamental properties of pseudo-
convex domains and plurisubharmonic functions are studied in the text of
Appendix I
372
Ph. Noverraz [545].
More recent articles on plurisubharmonic functions
and polar sets are S. Dineen [193,196], E. Ligocka [444], M. Esteves and C. Herves [231,232], S. Dineen and Ph. Noverraz [205,206], P. Lelong [438, 439,440), B. Aupetit [32],Ph. Noverraz [554,557) and C.O. Kiselman [388]. The next result on the equivalence of the various conditions is due to Ph. Noverraz [543,546]. subsets of cJJ J
J
S. Dineen [190].
He proved the Cartan-Thullen theorem for open
spaces and this was extended to JJ1m spaces by L. Gruman [289,290] was the first to give a complete
solution to the Levi problem in an infinite dimensional space. the solution to the
a
He used
problem in finite dimensions and an inductive
construction to show that pseudo-convex domains in separable Hilbert spaces are domains of existence of holomorphic functions.
The technique and
result of L. Gruman have influenced almost all later solutions to the Levi problem.
He also showed that a finitely open pseudo-convex subset of a
vector space over
C is the domain of existence of a G-holomorphic
function (see also S. Dineen [186,187), J. Kajiwara [365,366,367,368], and Y. Fujimoto [249]).
L. Gruman and C.O. Kiselman [291] then solved the
Levi problem on Banach spaces with a Schauder basis and Y. Hervier [329] extended this result to domains spread.
In [546] and [548] Ph. Noverraz
extended the solution of the Levi problem to Banach spaces with the bounded approximation property and proved, for these spaces, the following Oka-Weil theorems:
(i)
UC:U'
is
then
H(U')
if TO
U and dense in
hull of each compact subset of
U'
are pseudo-convex domains with
H(U)
if and only if the
U is contained in
pseudo-convex open set and the compact subset H(U)
hull then every holomorphic germ on
by holomorphic functions on
U.
U;
K of
(ii)
H(U') if
U is a
U is equal to its
K can be approximated on
K
Both (i) and (ii) were generalized to
domains spread over Frechet spaces and (jj J
4
spaces with finite dimension-
al Schauder decompositions by M. Schottenloher [640].
Ph. Noverraz [548]
and R. Pomes [583,584] then solved the Levi problem for J)JJspaces with a Schauder basis. The next important development is due to M. Schottenloher [636,640]. He combined regular classes, admissible coverings, surjective limits and a subtle but very crucial modification of L. Gruman's construction to solve the Levi problem for domains spread over hereditary Lindelof locally convex spaces with a finite dimensional Schauder decomposition.
This
373
Further developments
collection of spaces contains all Frechet spaces and all a Schauder basis.
jj:1 frL
spaces with
Particular cases of Schottenloher's result are given in
S. Dineen, Ph. Noverraz and M. Schottenloher [207].
M. Schottenloher [636,
640] and P. Berner [59,60] obtained, independently, the following result: if
= lim
E
+--
E CY.
is an open surjective limit and every pseudo-convex domain
CY.£A spread over E ,CY.£A, is a domain of holomorphy (resp. domain of existence) CY. then every pseudo-convex domain spread over E is a domain of holomorphy (resp. domain of existence). In [36], V. Aurich showed that domains of existence of meromorphic functions in Banach spaces with Schauder bases are domains of existence of holomorphic functions. In [154], J.F. Colombeau and J. Mujica solved the Levi problem for open subsets of
JJ J 11.
spaces.
They reduced the Levi problem on
.'j)
J tl..
spaces to the Levi problem on a Hilbert space, where L. Gruman's result applied, by using surjective limits and by combining the fact, noticed by previous authors, that any open subset of a
:iJ J 11. space is also open with
respect to a weaker semi-metrizable locally convex topology with Grothendieck's result [286,288] that for any sequence of neighbourhoods of O,(Uj)j in a
OF
f}AjU j
space there exists a sequence of scalars
(A.).
J J
such that
is also a neighbourhood of zero (see also corollary 2.30).
approach has been developed by J.F. Colombeau and J.
~\ujica
This
[156] in their
study of Hahn-Banach extension theorems and convolution equations. In [506], J. Mujica solves the Levi problem for domains in E
(E',T O)'
a separable Frechet space with the approximation property by using
topological methodi.
Mujica also proves in [506] that a holomorphically
convex domain in (E',T )' E a separable Fr~chet space, is the domain of O existence of a holomorphic function and this result was extended, using quite different methods, by M. Valdivia [691] to the case where arbitrary Frechet space.
E
is an
M. Valdivia obtains a number of interpolation
theorems for vector valued holomorphic functions in [691].
See also M.
Schottenloher [636]. This completes our survey of the Levi problem and the Cartan-Thullen theorem in infinite dimensional spaces.
Our analysis has hopefully shown
their central role in infinite dimensional holomorphy and their importance
Appendix I
374
in motivating new ideas and concepts.
This direction of research still
contains many open problems, e.g. the Levi problem has not been solved and no Cartan-Thullen theorem exists for arbitrary domains in separable Banach spaces.
Indeed the reader will no doubt have observed that all known pos-
itive results on the Levi problem involve an approximation property assumption and this excludes certain separable Banach spaces.
Further
references for the above topics are J. Horvath [349], W. Bogdanowicz [77] D. Burghela and A. Duma [110], E. Ligocka and J. Siciak [446], E. Ligocka [444,445], M. Herve [325,326], J. Bochnak and J. Siciak [75], C.E. Rickart [606], S. Baryton [54], I.G. Craw [170], S. Dineen [193], G. Coeure [132, 133,134], G. Katz [372], J. Kajiwara [365], V. Aurich [34,37], Y. Hervier [330], L.A. de Moraes [495,496,497], A. Bayoumi [55], M.G. Zaidenberg [719], S.J. Greenfield [279] and Y. Fujimoto [249]. In finite dimensions fundamental solutions of the
a
operator can be
obtained from the potential kernel, i.e. from a fundamental solution of the Laplacian.
L. Gross [284] (see also P. L~vy [442]) has studied infinite
dimensional generalizations of the potential kernel and found, because of the absence of a translation invariant (i.e. Lebesgue) measure on an infinite dimensional locally convex space, that the natural setting for finding fundamental solutions of the Laplacian was an abstract Wiener space with its associated Gaussian measure. Wiener space if
j
B is, via
(j,H,B)
H is a separable Hilbert space,
is a continuous injection of of
A triple
j,
is called an abstract B is a Banach space,
H onto a dense subspace of
a "measurable" norm on
H
B and the norm
(if for instance,
is a Hilbert-Schmidt operator with non-zero eigenvalues, then
is an abstract Wiener space). leads to a true measure on C.J. Henrich
The canonical Gaussian "measure" on
B for any abstract Wiener space
[322] was the first to investigate the
an infinite dimensional setting.
d
H=B
and
(j,H,H) H
(j,H,B). equation in
His approach was influenced by the work
of L. Gross [284] on the infinite dimensional Laplacian, by H. Skoda's research [662] on the finite dimensional L. Hormander [346] on C.J. Henrich's
L2
d
equation and by the work of
estimates for partial differential operators.
work is very fundamental, quite delicate (even the state-
ment of the equation and the interpretation of the solution necessitate a careful examination) and his ideas have influenced later developments. main result is the following:
His
Further developments if
H is a separable
form on
Hilbert space and
375
w
is an
(0,1)
H which factors through an abstract Wiener space
(*)
as a closed form of polynomial growth, then there exists a . tJ
,)(P
q>
3a
function of polynomial growth on
H,a,
= w.
The condition on abstract Wiener space ial growth on
w in
means the following:
(*)
(j,H,B),
a
a
closed
(0,1)
there exists an
form
w of polynom-
B such that the following diagram commutes
Equivalently we may say that dense subspace of
H.
(*)
solution to the
d
a
is a solution to the
equation on a
In [421], B. Lascar shows that Henrich's
can be extended to the whole space (i.e. to H)
wara.
such that
solution
as a distributional
equation.
A summary of the work of C.J. Henrich is given in [364] by J. KajiThe formula for Henrich's solution is very technical, mainly because
Gaussian measures are not invariant under translation and this leads to complicated terms when differentiating under the integral sign. In [187], S. Dineen used transfinite induction and sheaf cohomology to show that each infinitely G~teaux differentiable closed a finitely open pseudo-convex subset image by
a
Q
(0,1)
form on
of a complex vector space is the
of an infinitely G~teaux differentiable function on
Q.
In his study of the representation of distributions by boundary values of holomorphic functions, D. Vogt [701] encountered the vector valued problem and discovered owing result [701].
If
DN E
spaces is a
(definition 5.38).
ll111 space,
then the following conditions
are equivalent: 1)
d
He proved the foll-
each E-valued distribution of compact support in
R may be
376
Appendix I represented as the boundary value of an element of H(C\R;E) ,
a : _R:;. (R 2 ;E) ->-t,CR2 ;E) -
(2) the mapping
Efo,
(3)
is a
ON
00
00
is surjective,
space.
A. Rapp [601,602] solved the equation
3a =
w
on a convex open subset
of a Banach space with regular boundary when the closed form
w is of
sufficiently slow growth near the boundary and E. Ligocka [444,445] obtain-
~l
ed a solution for
functions of bounded support on -a Banach space.
Both used straightforward generalizations of the finite dimensional method. Next, P. Raboin made a number of important contributions by returning to the approach of C.J. Henrich he defined the space
L2
q
of
and using Gaussian measures. (O,q)
integrable with respect to the Gaussian measure Hilbert space
H
In [587,589]
differential forms which are square ~
on the separable
and showed that the restriction
a closed operator with dense range.
T of d to L2 was q q After obtaining an integral represen-
tation for the adjoint of
Tq and establishing a priori estimates (in the manner of L. Hormander [346] for the finite dimensional case) he proved L2 that each closed form in L2 q+l was the 3 image of a member of q He proved that each _-e closed form in L2 was the image of an element 1 of L2 whose restriction to a certain dense subspace of H was a "...e, 00" 00
function. the
:i
In [589], Raboin showed the existence of a
problem for
£,00
closed
(O,l)
".e 1 "
solution to
forms, bounded on bounded sets, and
extended this result in [593], (see also
[590,591,592]), to pseudo-convex
domains in a Hilbert space by using a generalized Cauchy integral formula for
..e,
00
functions.
In [137], G. Coeur~ gives an example of a ~l closed (0,1) the unit ball ~l
B of a Hilbert space which is not the image by
function on
form on
a of any
B.
The natural step from Hilbert spaces to nuclear spaces, suggested by C.J. Henrich
[322], was taken by P. Raboin in [588,590,591,592,593].
[593], he proved that any ~oo
closed
(0,1)
form, satisfying a modest
technical condition on a pseudo-convex open subset
Q
of a £)1-11 space
In
377
Further developments with a basis was the image by
1
of a
"3
1.0
fUJilction on
J.F. Colombeau and B. Perrot prove that every
JJJT1
function on
space
E
a
is the image by
remark by P. Kree in
§6.G
to pseudo-convex domains in by D. Nosske [531]).
of a
closed
(0,1) E
form on a
(see also the
and in [166] they extend this result
of [418] ) E
~oo
In [164],
Q•
.too
(this result was also found, independently,
The initial version of J.F. Colombeau's and
B. Perrot's solution to the
a
problem [166] was considerably simplified
by a result on hypoellipticity (due to P. Mazet [480]) which yields, as a particular case, the following: "~oo solution to the
Any G£teaux
a problem which is locally bounded
is a (Frechet) ~oo solution. Recently, R. Meise and D. Vogt [488], have shown that the solvability
a problem on a nuclear Frechet space E implies that E has
of the property
DN
(definition 5.38).
Applications of the infinite dimensional
a operator to natural
Fr:chet algebras are given in B. Kramm [398] and to convolution operators by J.F. Colombeau, R. Gay and B. Perrot in [148].
Application of the , operator to the Cousin I problem are discussed below.
3
SHEAF THEORY Sheaf theory and sheaf cohomology play an important role in several complex variable theory and it is probable, see for instance B. Kramm [398], that the same remark will eventually apply to infinite dimensional holomorphy.
Of key importance for finite dimensional holomorphy are theorems A Theorem B states that HP ex, 'f) = 0 for any
and B of H. Cartan [115]. p?:
1
and any coherent analytic sheaf
Theorem B can be used to solve the
a
f
on the Stein manifold
x.
problem and to resolve the Cousin I
problem (also called the additive Cousin problem) on holomorphically convex domains in
[no
Classically the Cousin I problem was to find a several
complex variable version of the Mittag-Lefflertheorem - which showed the existence of a meromorphic function in any domain of poles.
[
with preassigned
The several complex variables version sought to characterise within
the collection of principal parts on a domain rise to a meromorphic function on in [115].
X.
X in
(n
those which gave
This problem was solved by H. Cart an
378
Appendix I
In recent years, various authors (e.g. L. Hormander [347], C.E. Rickart [605] and P. Raboin [588]) have assigned the terminology "Cousin I problem" to a more general collection of problems of which the following is typical: given a covering
(Ui)isI
locally convex space
E,
of a domain
X,
spread over a
and
for all
i, j s I
such that
o on
h .. +h .. 1J
J1
for all
i,j
hi s H(U i ) in I?
U. 1
n
J
o on
h .. +h., +h . 1J JK k 1
and
U.
(**)
and k in I,does there exist a family (hi)isI' such that h.-h. h .. on U. n U. for all i and J
1
1J
1
J
v
Using Cech cohomology we see that (**) has a solution for any set of data {U.,h .. } 1
on
X.
Hl(X,~)
if
1J
= 0
where
~ denotes the sheaf of holomorphic germs
It is easy to show that a generalised Mittag-Leffler theorem is
valid on
X whenever
Hl(X,~)
= O.
Banach algebra considerations motivated the first examples of sheaf cohomology with values in a sheaf of holomorphic germs in infinitely many variables. where
HP(~'~AI) = 0
In [12], R. Arens proved that
A is a Banach algebra with continuous dual
and where {JI AI
A'
p~l
for any
and spectrum
is the sheaf of weak* holomorphic germs on
AI.
!
He appHed
this result to show {xEA, x invertibl~exp(x); xsA} (see also R. Arens [13] and H. Royden [610]). In [605], C.E. Rickart proved that any polynomially convex compact subset
HP(K,0) K of
[A,
o for any
and
p~l
A arbitrary.
C.E.
Rickart [605] also states and solves a Cousin I problem on the set
K and
applies it to prove the Rossi local maximum modulus principle for Banach algebras. P. Silici shows in [659] that theorems
A and
B are valid for
379
Further developments
compact polydiscs in proved that
[A.
Hl(U'G) '" 0
By using transfinite induction, S. Dineen [187] for any finitely open pseudo-convex domain
U
in a complex vector space, where i9 G is the sheaf of G~teaux holomorphic germs, and used this result to solve the Levi problem and the a problem _ .Doo for Gateaux holomorphic and uateaux ~ functions. J. Kajiwara [368] ~
extended this result to the higher cohomology groups on finitely open pseudo-convex domains in projective space (see also Y. Fujimoto [249]).
In
[192], S. Dineen showed that Cousin I is not solvable, and hence
Hl(U,~)
F0
a problem is not solvable, for any domain U in a
and the
locally convex space which does not admit a continuous norm and in [35]
v.
Aurich proved that a given family of principal parts on a Stein manifold CA, A arbitrary, gives rise to a meromorphic function if and
spread over
only if the principal parts all factor through some
[no
The next development is due to P. Raboin [588] who proved, using his solution to the
8
problem, the following Cousin I result;
pseudo convex domain in a Frechet nuclear space {Q.,g .. } . . 1
1J 1, J
is a set of Cousin I data on
balanced subset
K of
E
if
Q is a
E with a basis and
then for each convex compact
there exists a family
{f i E H(Qi"EK)}i such that g .. = f.-f. on Q.()Q.()E for all i and j. (E K is the Banach K 1J 1 J 1 J space with closed unit ball K and each fi is holomorphic with respect to the topology of
EK.) In [593], P. Raboin proved that HI( U,0) = 0 for any pseudo-convex domain U in a tJJ-11 space with a basis. His proof
involved a solution of the that
IJ fl1
of unity.
a problem, the Oka-Weil theorem and the fact
spaces are hereditary Lindelof spaces and admit,.e,oo This result was extended to arbitrary
JJJll
partitions
spaces by J.F.
Colombeau and B. Perrot [164,166]. Theorems A and B of H. Cartan have been extended to vector valued holomorphic functions on a finite dimensional space by L. Bungart [109]. This completes our discussion of conditions (a),(b), ... ,(f) for infinite dimensional spaces.
DIFFERENTIAL EQUATIONS We now discuss convolution operators and partial differential operators on spaces of holomorphic functions over locally convex spaces. As this subject forms part of a book in preparation by J.F. COlombeau, our
Appendix I
380
presentation will be brief and concentrate mainly on the role played by this topic in the general development of infinite dimensional holomorphy. C.P. Gupta [295] was the first to consider convolution operators on spaces of holomorphic functions over locally convex spaces and his approach influenced many later workers in this area.
The main finite dimensional
considerations of C.P. Gupta were the results and techniques of B. Malgrange [448] and A. Martineau [452]. A simplified description of the basic approach used by C.P. Gupta goes as follows.
Given
A
a
locally convex translation invariant space of
holomorphic functions on the locally convex space ator on
E,
a convolution oper-
A is defined as a continuous linear operator from
which commutes with all translations. operator has the form
a
I:=o where
For
14
H(II:)
=
4
into itself
each convolution
n
nl
1
lim sup I a In < n
n--
00
The Borel transform establishes a one-to-one correspondence between convolution operators on ~,
the elements of ~,
and a space of holomorphic
functions of exponential type on E'. The existence and approximation problem for convolution operators is then transposed and solved as a division problem for holomorphic functions on
E'.
C.P. Gupta's [295] investigation of convolution operators on Banach spaces led him to bounded type on
~b(E),
E,
the space of holomorphic functions of nuclear
and to the correspondence
showed that every convolution operator on
HNb(E)S
HNb(E)
= Exp(E }' He S was surjective and that
solutions of the associated homogeneous equation could be approximated by exponential polynomial solutions.
Extensions of this method to more general
classes of locally convex spaces and to other collections of holomorphic functions are given in C.P. Gupta [296,297], L. Nachbin [511], P.J. Boland [79,80,81,82], M.C. Matos [458,463,464,467], S. Dineen [177], P.J. Boland and S. Dineen [88], T.A.W. Dwyer [218,221,222,223,225], P. Berner [62], D. Pisanelli [580,581], J.F. Colombeau and M.C. Matos [150,151], J.F. Colombeau and B. Perrot [163,167], F. Colombeau, T.A.W. Dwyer and B. Perrot [147], and J.F. Colombeau and J. Mujica [156].
381
Further developments
A different approach is taken by T.A. Dwyer [214,215,216,219] (see also O. Bonnin [94]) in studying partial differential operators on holomorphic Fock spaces of Hilbert-Schmidt type. ~p(E)
on a Hilbert space
E
He defines the Fock spaces
(and afterwards on countably Hilbert spaces
and other classes of locally convex spaces, see also J. Rzewuski [621,622]) and shows that
II Pfll P
~
II Pmil p
partial differential operator
II flip
.
for any
I:=o
P(D)
in
f
Pn(D).
J- P (E)
Using this inequality
Dwyer showed that all such partial differential operators map ~p(E)
and any
J
p
(E)
onto
and generalised a number of finite dimensional results (see F.
Treves [686], chapter 9).
Notable aspects of Dwyer's work, see the refer-
ences cited above and [224,226,227], are his concrete representation of convolution operators by means of
L2
(Volterra) kernels, etc. and his
recognition of a relationship between certain abstract differential equations in locally convex spaces and problems in control theory, analytic bilinear realizations, quantum field theory, etc. (see also J.F. Colombeau and B. Perrot [158,162], J.F. Colombeau [145], P. Kree [401,410,417] and P. Kree and R. Raczka [419]). The long term relevence of convolution operators in infinitely many variables may well depend on this kind of recognition and insight. The most recent developments in this general direction are due to J.F. Colombeau, R. Gay and B. Perrot [148].
They prove, using a prepar-
ation theorem for holomorphic functions on a Banach space due to J.P. Ramis
= j'(~)
[598] (see below), that
f~'(~)
holomorphic function
on a connected domain
nuclear space
p
a
HM(E)
in a quasi-complete dual ~oo
problem on :Jj JY'l. spaces to prove the following:
is a convolution operator on E
~
E and apply this result together with the existence of
solutions of the T
f
for any non-zero Mackey (or Silva)
then any solution
transform of an element
f
U of
Exp(E')
with characteristic function
of the equation
t:'
(E)
for which
Tf pU
=0 =
is the Borel
o.
The finite
dimensional analogues of these results are due to L. Schwartz [647] and R. Gay [254] respectively. The theory of convolution operators drew attention to the role of nuclear polynomials in the general theory of holomorphic functions in infinitely many variables and provided the first examples of a function space representation of infinite dimensional analytic functionals.
The
if
Appendix I
382
appearance of nuclearity motivated L. Nachbin [508,509] to introduce the concept of holomorphy type as a means of investigating holomorphic functions whose derivatives pertained to a certain class of polynomials (e.g. compact, Hilbert-Schmidt, nuclear, etc.) and whose Taylor series expansion satisfied growth conditions relative to the canonical semi-norms on the underlying spaces of homogeneous polynomials.
The theory of holo-
morphy types has been developed in various directions by L. Nachbin [508, 509,511,512], S. Dineen [177], R.M. Aron [15,16,17,19], T.A.W. Dwyer [214, 216,221,222,223], P.J. Boland [78,80,81], S.B. Chae [119,120] and L.A. de Moraes [495,496,497], and led, eventually, through the work of Boland, to the theory of ho1omorphic functions on nuclear spaces as outlined in chapters 1,3,5 and 6. The Borel transform and the correspondence between analytic functionals on
H(E)
and holomorphic functions of exponential type on
E'
were
almost totally developed within the framework of convolution operators as outlined above (see the references listed previously and also T.A.W. Dwyer [217,220]), and although in this text we have more or less exclusively preferred the representation of analytic functionals by holomorphic germs the motivations and guidelines arising from the exponential representation were always suggestive and significant. ANALYTIC GEOMETRY The first comprehensive treatments of analytic sets in infinite dimemsions are due to P.J. Ramis [594] and G. Ruget [613], both of whom worked only in Banach spaces but were aware that many of their results and techniques extended to locally convex spaces.
Most of the other important
developments in this area are due to P. Mazet [479].
As in the finite
dimensional case (see for instance M. Herve [324]) the local theory is first developed by studying the ideal structure of the commutative ring ~(E)
space
(the space of holomorphic germs at the origin in the locally convex E), and then applied to obtain global results.
The ring 0CE}
an integral domain and a local ring but is Noetherian if and only if finite dimensional.
Since the Noetherian property of 19([n)
is E is
plays a
crucial role in the finite dimensional theory, new methods in commutative algebra had to be developed and these appear to be of independent interest.
383
Further developments
Two of the key results in both the finite and infinite dimensional theories are the Weierstrass Factorization and Preparation Theorems. (J.P. Ramis [594] and P. Mazet [479]).
Weierstrass Factorization Theorem If
g f. 0 £ \9(E)
then there exists a decomposition of
that the restriction of
g
to
Ce
there exists a unique polynomial such that
~(E)
p f. 0
has order r
of degree
uM and so
5.71
-zl-zw
n n
then
apply corollary 5.35.
(tRf)(z) w
la b
In particular, we have If
< l.
this shows
mEN(N)
I mc:N(N)
a z m
m
a ( _z_ )m m l-zw
z ( _ n _ ) "" 1-z w n n n=l
Show also that the mapping
E HHY (U)
show that
where
427
Notes on some exercises
z
z
-+
---
EO
H(U).
l-zw For further details consult P. Boland and S. Dineen [91]. 5.74
See R. Soraggi [669].
A quotient mapping is an open mapping.
Show that the canonical mapping from
H(OE)
using the definition of inductive limit.
onto
If
E
HCOF)
is continuous by
is fully nuclear this shows
that HCO ) is regular if HCO ) is regular and transferring this to the F E dual space we obtain the following: if ~\ is a closed subspace of a fully nuclear space M2 and , o =, o,b on H(M ). In certain cases, for instance when l can replace 'o,b by '0·
H(M ) then , =, on o 2 o,b is a Frechet space, one
M2
5.75
See J.F. Colombeau and R. Meise [152].
5.76
See P.J. Boland and S. Dineen [91].
and hence
cfr>cnE)"o)
and
E is an
CCcPCnE)"o)S)S ~ ( asoes nos espacos f unClonalS ana I""ltlCOS. Bol. Soc. Mat. Sao Paulo, ,5 (19501, 1952, p.1-58.
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[N.
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This Page Intentionally Left Blank
INDEX
A-nuclear space ................................ .
226, 425
A-nuclear space, reflexive ..................... .
229
Abel's theorem
102
Absolute basis
228, 229
Absolute decomposition
114 374
Abstract Wiener space
368, 372
Admissable coverings
2
Algebraic dual Algebraic hyperplane ........................... .
215
Analytic bilinear realization .................. .
381
156, 236, 292, 296, 334, 361
Analytic functional ............... .
363, 381, 382,420
Analytic set
383
Analytic set, codimension of .................... .
386
Analytic set, finitely defined .................. .
383
Analytic set, germ of ........................... .
384
Analytic set, irreducible
384, 386
Analytic set, principal
383
Anticommutative forms
428
Approximation property
40,
46, 139, 209, 289, 328
333, 359, 369, 370, 371, 404
Ascoli's theorem ....................... . Associated barrelled topology ............... .
131, 155, 398, 435, 421 112, 300
Associated sequence
337
Associated topology
74, 110, 146, 153 481
482
Index
B-continuous function .................................. .
411
B-nuclear space ...............•..........•..............
267
B, property
430
Baire space
411
Baire theorem ....••....................................
68, 399
Banach-Dieudonne theorem
412
Banach-Lie group
393
Banach-Stone theorem ................•.................
163
Barrelled space ...........•.•......................... Basis
94, 183, 404
Basis, absolute ..................................... . Basis, equicontinuous
(equi~Schauder)
............... .
24, I 12, 400 218, 229 219, 404
Basis, monotone
209
Basis, Schauder
218, 229, 278, 293, 369, 404
Basis, shrinking ............................... .
293
Basis , unconditional ........................... .
183, 289, 404
Biholomorphic mapping .......................... .
205, 206, 384, 388
Bilinear mapping
2, 19O, 406
Borel measurable function ...•...................
380, 381, 382
Borel transform ..•..............................
31, 137, 249
Bornological space .....•........................
16, I I I , 400
Bornological space, DF .•............................
131
Boundary values (of holomorphic function) ....•.......
375
Boundedly retractive inductive limit ..... '" ....... . Bounding set
173, 202, 203, 368
Boundedness, radius of ....•.........................
166, 206
c* algebra ......................................... .
390
Calculus of variations ......•.......................
101
Caratheodory metric ................................ .
392
Cartan domain
389
Cartan factor
312, 357
390
Cartan-Thullen theorem .................•..........
365, 368, 370, 372
Category ............•.................•...........
415
Category, first---subset ....................... .
207
Category of locally convex spaces ..•..............
16, 400
Category of topological spaces ................... .
16, 54, 399
483
Index
subset ..................•••..••
43
Cauchy estimates ..................•................
90, 30 I, 422
Cauchy-Hadamard formula .........•..................
165, 338
Category, second -
Cauchy inequalities •..................••......•.•..
57, 408
Cauchy integral formula ................•.•.........
237, 376, 408
Cauchy - Riemann equations ...................•....•
54, 103
Cesaro sums ....................................•...
196
Closed forms .......••••..............•.............
375
Codimension (of analytic set) ....................•.
386
Coherence (of Taylor series expansions) .•.••........
90, 301
Coherent Sheaves ..•••...........................•.•
377
Compact mapping ...•.....................•.........•
93, 152
Compact-open topology ....•...........•.......•.....
23, 71, 399
Compact operator
430
Cousin I problem
104, 378
Control theory
381
Convergence, Mackey---criterion ....•........•....•.
62
Convergence, pointwise ............................ .
96, 148, 399
357 Convergence, strict Mackey-criterion ••.•.......... 81, 97,149,281, 321, 325 Convergent, very strongly ........... .
Convergent, very weakly .•..................
82,
97
Convolution operators ...•............•..••.
104, 380, 418, 421
Curve of quickest descent ................. .
101
d
problem
DF ....................•................... DFC DFM DFN DFS f
10
10
••••••
~
•
'"
...........................
.
decomposition, absolute •............•..•.• decomposition, equi-Schauder ............. . '7
decomposition, ,/_absolute .......••......... decomposition, ~schauder •............•....
365, 371 , 374, 379 18, 131, 147, 307, 403 419 14 17 IS
114 114 114 I 14
Schauder .............•.....
114, 294
decomposition, shrinking .................. .
114, 147
determining manifold ...•..........••..•.••.
414
determining set direct image theorem •.........•..........
211,319, 425
decomposition,
388
484
Index
distinguished Frechet space distributional solution
of
25,
34, 357
3 ............ .
375
division theorem ....................•......
380, 384, 421
Dixmier-Ng theorem ........................ .
417
domain of existence ........................ .
365, 367, 372
domain of
holomorphy ..................... .
365
domain, polynomially convex ............... .
213, 359, 369
domain, pseudo-convex ..................... .
64, 335, 365
domain spread ............................. .
367
dominated
norm CDN) space ...... .
262, 288, 334, 375, 377, 429 -45, 293, 413, 423
Dunford-Pettis property .............. . eigenvalues
374, 394, 413
envelope of holomorphy ....................... .
366, 368
evaluation mapping ........................... .
329
exponent sequence ............................. .
336
exponent sequence, nuclear .................... .
336
exponent sequence, stable ..................... .
336
exponential polynomial solutions .............. .
380
exponential type, functions of ................ .
.IS 1, 156, 420, 421
exponential type, functions of nuclear ........ .
152
extreme points ............... .
161, 204, 205, 211, 405 105
factorization, global ........................ . 11,
factorization lemma
63,
98
factorization properties ..................... .
367
factorization theorem ...... '" ............... .
296
finitely open topology ....................... . finitely polynomially convex domain ........... .
16,
53,
92, 379, 411 369
Finsler metric ............................... .
393, 422
fixed point theorem .......................... .
206, 369, 422
Fock space ................................... .
381
Fre'chet space ................................ .
12, 400
Fredholm operator ............................ .
387
fully nuclear space ........................... .
33, 139, 229
fully nuclear space, with basis .............. .
229
functional calculus
356
Index
485
G holomorphic function .............................................
54
Gateaux holomorphic function .......................................
54
Gaussian measures .................................... .
215, 374
geometry of Banach spaces ............................ .
159
geometric ideal ..................................... .
385
germ, holomorphic ................................... .
84, 250
germ, hypoanalytic .................................. .
91, 255
germ, of analytic set ............................... .
384
germ, nuclear holomorphic ........................... .
138
Grothendieck-Pietsch criterion ....................... .
222, 431
Hahn-Banach theorem
401
Hahn-Banach theorem, holomorphic ....... .
202, 215, 296, 418
Hartogs' theorem ........................ .
54,
59, 103, 409, 415
hemicompact space ....................... .
397, 419
holomorphic completion ................... .
203, 215, 371
holomorphic function
57
holomorphic function of nuclear bounded type .... .
386
holomorphic function of nuclear type ........... .
156, 421
holomorphic,
G ----- function ................. .
54
holomorphic
germ .............................. .
84, 250 136
holomorphic, nuclear ----- germs holomorphic vector field ....................... .
393
holomorphically convex domain ................... .
311, 365, 409
holomorphy type ................................ .
51, 382
homogeneous domain ............................. .
389
homogenous polynomial .......................... .
3
homogeneous subspace ........................... .
196
hypoanalytic function .......................... .
60, 319
hypoanalytic germ
91, 255
hypocontinuous function ........................ . hypo continuous
homogeneous polynomial ............ .
397 13
ideal, maximal
369
ideal, geometric ................................... .
385
induction .................. . induction, transfinite ................ .
2, 13, 28, 39, 194, 266, 343 112, 379, 420
486
Index
inductive limit, boundedly retractive .................... .
312, 357
.....................
16, 400
inductive limit,
locally convex --topology
..... .......... .........
86,
97, 141, 253, 304
inductive limit, strict ...................................
17,
34,
inductive limit, regular
~
~
98, 278, 405
inductive limit topology
16, 399
inductive tensor product
406
infinite matrix
263
infrabarrelled space ....................................... .
401
intersection theorem ...................................... .
368
invariant metric
393
iteration method
422
irreducible domain ..........................•..............
389
irreducible analytic set ...•...............................
384, 386
163, 211, 390, 422
J* algebra J*
394
triple
390
Jordan algebra ...................... . k
14,81, 104; 332,397,415,419
space ...................... .
Kelley space
III
kernel theorem................................
334, 363
Krein-Milman theorem .....•...... ..............
211,406
Levi problem...........................
68, 104, 204, 214, 365, 372
Lie algebra ............................
394 393
Lie group Lifting theorems ...................... .
157
limited set ........................... .
212
Lindelof space .... .••.•... .•.•.. Liouville's theorem ............ .
45,68,69,95, 290, 372, 379, 397 110, I I I , 408
local boundedness ..............•.........
413
local connectedness .............•......•
301
local maximum modulus principle ......... .
378
local (sheaf) topology .................. .
75, 308
local uniform topology ..•...............
393
locally bounded function ......••......
10,58, 77, 104, 199, 258, 290
locally m convex algebra ........•....
98, 99, 355, 417
487
Index
M closure topology ....................................... .
15
Mackey - Arens theorem ................................... .
402
Mackey continuous ........................................ .
14
Mackey convergence criterion .............................. .
62
Mackey convergent sequence ............................... .
14
Mackey holomorphic function (Silva) ...................... .
61
Mackey space ......................... .
35, 291, 402, 425
Mackey , strict-convergence criterion
357
topology ........................ .
35
mapping, biholomorphic ........................... .
384, 388
mapping, bilinear ................................ .
2, 184, 406
mapping, compact ................................. .
93, 152
mapping, diagonal ................................ .
3
mapping, holomorphic ............................. .
57
mapping, n-linear mapping,
nuclear
mapping,
symmetric
21, 402 2
maximum modulus theorem .......................... .
408
meromorphic function .............................. .
104, 373, 377, 386
Mittag-Leffler theorem ........................... .
103, 377
Mobius transformation ............................. .
211, 391, 422
Modular hull ...................................... .
223, 230
Modularly decreasing set ......................... .
230~ 288
Montel space ..................................... .
14, 402
Montel theorem
155, 408, 416
Morera theorem
102
multiplicative linear functional ................. , multiplicative
polar
106, 290, 366 290
Noetherian ring
382
Normal decomposition ............................ .
385
normal mapping
94
normal topology
364
nowhere dense set ............................... .
177
nuclear, dual ---space nuclear
exponent sequence
nuclear, fully ---- space nuclear function of --- exponential type ......... .
21 336 33, 229 152
488
Index
nuclear nuclear
space ....................... .
" mapping
157, 363 21, 402
nuclear polynomial nuclear,
s
21 space .................. .
56
nuclear
sequence space
nuclear
space ....................................... .
21, 403
222
nuclearly entire functions ........................... . Nullstellensatz
136 384, 385
numbering function ................................... .
340
numerical range ...................................... .
205, 211, 422
Oka-Weil theorem ..................................... .
311, 370, 379
Open mapping theorem ............................... .
278, 307, 405, 430
Orlicz spaces ...................................... .
214
paracompact spaces ................................. .
45,
95
partial differential operator ..................... .
374, 379
Patil problem ..................................... .
424
plurisubharmonic functions ........................ .
170, 366
Poincare metric ................................... .
391
polar set ......................................... .
215, 366
polar, multiplicative ............................. .
290
polarization formula .............................. .
4, 320
polydisc .......................................... .
230
polynomial ........................................ .
3
polynomial, bounded on equicontinuous sets ........ .
32
polynomial, continuous
10
polynomial growth ................................. .
375
polynomial, hypocontinuous ........................ .
13
polynomial,
Mackey continuous .................... .
14
polynomial,
homogeneous ......................... .
3
polynomial,
nuclear ............................. .
21
polynomial,
weakly compact ...................... .
45
polynomially convex domain ........................ .
213, 359, 369
pointwise convergence .............................. .
148, 399
ported topology ................................... . ported
24,
72
semi-norm ................................. .
72
power series space ................................ .
268, 289, 336
489
Index
3B3
freparation theorem
377
principal parts product,
e:
product, tensor
328, 407 I, 49, 328, 334, 406 406
projective tensor product ..................... . proper mapping
387
property (B)
430
property (S) pseudo-convex topologies ...................... . Q
family
34,
62 411 110
quantum field theory
157, 295, 3BI
quasi-normable space .•..........•..............
133, 313, 358
radius of boundedness ......................... .
166, 206
radius of pointwise convergence ............... .
166
radius of uniform convergence ................. .
103, 166
Radon- Nikodym Property ....................... .
178
rapidly decreasing sequence ................... . ramified coverings
262, 291, 356, 429 385
reflexive space
401
regular classes
368, 372
regular inductive limit ............... .
86, 97, 141, 253, .304
regular point of analytic set ......... .
384
Reinhardt set
230 , 240, 359
Remmert graph theorem .......•..........
388
removable singularities ............... .
\03
residue theorem ....................... .
102
resolvent function .................... .
211
Riemann mapping theorem ......•.........
389
Rotund Banach space ....•....•..........
161
Rudin-Carleson interpolation theorem .......•....
424
Runge's theorem ............................... . Russo-Dye theorem
104, 370
J- absolute decomposition
4- Schauder decomposition
163 114 114
Index
490
Scftauder basis ....................... .
218, 229, 278, 293, 404
Schauder decomposition .............. .
114, 294
S'chwarz lemma ....................... .
161, 211, 391, 422
Schwarz-Pick system ..................... .
392
Schwarz-Pick inequality (condition) ...... .
392, 422 34, 152, 402, 421
Scftwartz space semi-Montel space ....................... .
14, 402
semi-Reflexive space .................... .
142, 259, 401
separately holomorpftic .................. .
54,
59, 148, 409, 415
sequence space,
221
sequence space, nuclear ................. .
222
sequential compactness
177
sequential convergence
178, 207, 212 377
sheaf cohomology sheaf (local) topology .................. .
75, 308
shrinking decomposition ................. .
114, 147
Silva holomorphic function .............. .
61 230
Solid set ............................... . space, barrelled
24, 112, 400
space, bornological ..................... .
16, III, 400
space, dispersed ........................ .
46
space, distinguished Fr~chet ............. . space, dominated norm (DN) .............. .
25,
34, 357
262, 288, 334, 375, 376
397, 419 space, hemicompact ...................... . 14, 81, 81, 104, 332, 397, 415, 419 space, k ......................
space, Kelley
.............................
space, Lindelof
.................
45, 68, 69,
III
95, 290, 365, 372, 397 35, 291, 402, 425
space, Mackey
...................
space, Montel
...................
14, Lf02
space, nuclear ................... space, paracompact ..............
21, 403
............ ................. semi-Montel .............. semi-reflexive ........... superinductive ........... ultra bornological .......
space, quasinormable space, Schwartz space, space, space, space,
45,
95
133, 313, 358 34, 152, 402 14, 402 142, 259, 401 15,
68
24, 111, 400
491
Index
~J?ace
, w
105 204, 211, 422
spectral radius
390
spectral decomposition theorem ........... .
357
strict Mackey convergence criterion .......... . strict inductive limit ........................ .
17,
34,
98, 278, 405
strictly compact set ......................... .
99, 418
strictly c convex Banach space ............... .
161 161
strictly convex Banach space ................. .
22, 401
strong topology subharmonic function
422
superinductive space
68, 415
15,
316, 362, 367, 373
surjective limit surjective limit, open ....................... .
316
surjective limit, compact .................... .
316
surjective limit, directed ................... .
358
symmetric domain
389 296, 354, 363
symmetric tensor algebra
2
symmetrization operator T.S.
completeness
128, 148
Taylor series expansion ..................... .
54, 120
tensor products ................ '" ... .
1,49,328, 334, 406, 413
topology,associated .................. .
110, 146, 153 23,
topology, compact open ............... . topology, finitely open .............. . topology,
16,
53,
71, 399
92, 379, 411
Kelley .................... .
III
topology, local (sheaf) ............... .
75, 308
topology, local uniform .............. .
393
topology, Mackey ..................... .
35
of pointwise convergence ........... .
96, 148, 399
topology of the M closure ...•..•..............
15
topology
topology, ported ..•....••.•..•................
24,
topology, strong
22, 401
topOlogy, TO topology, TJI topology, T
w
72
73 92 24,
72
492
Index
ultra Dornological spaces ............. .
24, III, 400
unconditional basis .................... .
183, 289, 404 13,
uniform boundedness principle ......... .
50
uniform convexity
171, 423
uniform factoring
319
unique factorization domain ........... .
383
unit vector basis ....•................. universally measurable
179, 221
very strongly convergent sequence ...... . very weakly convergent sequence ....... . Vitali's theorem ...................... . weak Asplund
space
414 81,
97, 149, 281, 321, 325 82,
97
155, 416 212
weak holomorphy ....................... .
414
weak* sequentially compact ............ .
178, 207, 212
weakly compactly generated Banach space ..... .
178, 207, 212
weak conditionally compact
179
Weierstrass Factorization theorem ........... .
383
Weierstrass Preparation theorem .............. .
383
weights
221, 263
