Israel Journal of Mathematics, Volume 2, Issue 2 (1964)

DOUBLE NORMALS OF CONVEX BODIES(1) BY

NICOLAAS H. KUIPER

ABSTRACT

In this paper we study the set of double normals of a solid convex body in E n (e.g. n-simplex). There are at least n double normals. The lengths form a set of measure zero in R for n < 3, not necessarily so for n > 3.

1. The problem. A bounded point in euclidean n-dimensional Its boundary is denoted by dB. p,q e dB. It is called a double vectors in E , (x-p)(q-p)~O

compact convex set B with at least one interior vector space E = E n will be called a convex body. A chord is a line segment [p, q] with end points normal in case, in terms of inner products of

and (x - q) (p - q) __>O for a l l x e B .

1. What can be said about the set of double normals of a convex body? In particular: 2. Must a convex body in E n admit at least n double normals?(2) 2. Examples. The polar coordinate values of a vector v ~ E are by definition r = ~/~- and co = v/r. The unit vector co is a point of the unit sphere S "-1 c E. The antipodal equivalence class z = n(co) = ( c o , - co} of the unit vector co is a point of the real projective ( n - 1 ) - s p a c e p , - 1 , of which the unit sphere S "-1 is covering space under the double covering n : S " - 1 . _ ) p , - t . I f co is a unit vector in the direction of a double normal [p,q] of B, then so is The set of such unit

co=(q-P)/[q-Pl

-co=(P-q)/[P-ql.

Received August 18, 1964. (1) a. Research partially supported by the N.S.F.b. Lecture in the conference on differential geometry, June 1964, at Oberwohlfach, Germany. (2) This last question was problem 1 in "Unsolved problems in intuitive geometry" by V. Klee (Mimeographed notes, University af Washington (1960)). Some mathematicians, e.g. T. Ganea, have been aware that the solution of this problem is essentially contained in work of L. Lyusternik, L. Schnirelmann [5] and M. Morse. V. Klee and B. Griinbaum suggested that publication, in particular for the case that B is not known to be C2, is desirable anyhow.

For thetreatment of an analogous Coo-problemin a space with a Riemannian metric see Bos [3] 71

72

N.H. KUIPER

[June

vectors is therefore invariant under the antipodal map and it defines the unique set of double normal directions K ( B ) c F -1 which it covers. THEOREM 1. I f f : P n - t - ~ R is any real C2.function ( = with continuous second derivatives), then there exists a symmetric convex body B in E" with centre O, for which the set of double normal directions is

K(B) = K ( f ) c P " - t , where K ( f ) is the critical set

r ( f ) = {z I z ~ e " - t, (df)z = 0). Proof. For t > O, t sufficiently small, the point set defined in terms of polar coordinates (r, co) by r = 1 + tf@@)) is a hypersurface OB,, boundary of a body B t in E . For t converging to zero this hypersurface converges to the unit sphere S"-t, and the continuous first and second derivatives are included in this convergence. By compactness of P " - t it follows that a > 0 exists such that the hypersurface aB, has at each point all normal curvatures positive, as has s " - t , and is therefore strictly convex. Because B, is symmetric with respect to 0 and strictly convex, every chord that connects two tangent points on parallel tangent hyperplanes, passes through 0. All double normal directions are then found from the equation (1)

dr = d[1 + ,f(Tr(¢o))] = 0,

or

d f = O.

Consequently K ( B , ) = K ( f ) and the theorem is proved. THE CONDITION C2-. A function f : R"--, R will be called C 2- in case it is C a (continuous first derivatives) and for any Xo in the domain there exist 6 and N such that (2)

If ( x + h) - f ( x ) - (dfL(h)[ < N.

I hl 2

for all x with [x - xo I < 6 and h < 6. (df)x is the derivative of f at x. It sends the vector h into (df)~(h). If a function f on a C°-m-manifold (like P=) has for any C°°-chart I 7(M),

where the summation extends over all components of each critical set in each level set. As ~(M) is finite, the conclusion is valid in case the summation extends over an infinite number of non-zero terms. Hence we may exclude this and assume that there are only a finite number of critical levels, and that each level set has a finite number of components. LEMMA. Let c be the only critical value o f f in the half open interval (b,e] ¢-R. Suppose fb = {zlf(z) f(z) and f(hl(z)) >=f(z) If z e f t - , , then h~ 1(z) e f t _ , . Suppose moreover h~ 1 z ~fb, then

f(h~lz) < f(z) - 2el. Consequently one finds that h~TM (f~_,) Cfb

~ integer and 2~ el > c - b

Or, as V ~ f b , (5)

h~(V) D h~ (fb) D f c - ,

for 2~tel > c -- b

Analogously

(6)

h~(f~+~)Dfe

for 213ei > e - c

If hz e f t + , - f ~ _ , then z e f t + , - - f b by (4). If moreover hz¢ hV', that is z ~ V', then f ( h z ) - f ( z ) ~ 2e hence zefc-,. Consequently for any point hzef¢+,:

DOUBLE NORMALS OF CONVEX BODIES

19641

either

hzef~_,,

then zef~_,;

or

hz e h V ' ,

then z~ V';

or

hzefc+~ -f~_~ and h z ~ h V ' - then zef~_~.

77

This implies

(7)

h(fc-, U V') Df¢+,

From (5), (6) and (7) follows:

fe c hPt(fc+,)c h~h(f¢_~ U V ' ) c h~h(h~V UV') and fe is covered by the p + q contractible open sets

and

h~hh~(V~)

for i = 1, ...,p

h~hh(Vj')

for j = 1,-.-, q

q.e.d.

If M is the real projective n - 1-space/~- 1, then it is known that y(M) = y(P"- 1) = n and so for a function f : P " - l ~ R we have: COROLLARY

y ( F , P " - ' ) >_-n. F

In particular if y(F,P ~-~) = 1 for each F, for example if each F is one point, then the total number of these components is greater or equal to n. 5. Application to the double normals problem. Combining the results of section 4 with those of section 3 we get: THEOREM 4. A convex body B in E" has at least n double normals. I f K ( B ) c p , - i consists of a finite number of components F each belonging to a set of double normals of constant length, then (8)

]~ ?(F, pn-,) >= n F

For example if B is an ellipsoid in E 3 with two equal axes, then K(B) ~ p2 consists of a point and a projective line and the left hand side in (8) is 1 + 2 which is >_3. PROBLEM. Let O be the family of all convex symmetric bodies B in euclidean 3-space E 3, that have all double normals in a plane. Is there an upper bound to the ratio between the largest and smallest width of B for B in O? How much is it? The ratio is max g(to) min g(to) in case r < g(to) defines B in polar coordinates r and to.

78

N.H. KUIPER

[June

6. On the length of the double normals of a convex body. THEOREM 5. For n > 4 there exists a convex body in E of non-constant width, such that every width (a real value) is attained as the length of some double normal. The body can be chosen such that there is an arc in p~-i consisting of directions of double normals connecting a minimal width double normal direction to a maximal width double normal direction. Proof. Whitney [7] has given examples of C n- 2-functions f on the n - 1-cube i n- 1 < Rn- t with values in I = {t [ 0 < t < 1}, such that d f is zero at each point of an (non rectifiable) arc connecting two points with different f-values. This function can be carried over to P~-1 by imbedding I ~- i in P~-1 and extending the function suitably (Whitney [8])(3). The construction of section 2 then gives the convex body required in the theorem.

PROBLEM. In Theorem 4 it remains open whether the set of all double normal directions (not decomposed in constant length parts) obeys an analogous relation If F represents a component of this set, is then again ~ r y ( F , P "-1) > n? A special problem in the same direction is: Is there on the two-sphere a C1-function f such that the set {x I (df ) x = 0} is an arc? F o r n = 3 we get a conclusion different from that in Theorem 5: TrIEOREM 6. The lengths of the double normals of a convex body B in E 3 form a set of measure zero in R. Proof. This mainly consists in an application of the theorem of A. P. Morse and Sard. As in section 3 we replace B by B" and then by B', and we denote the latter again by B . Let this symmetric body B be defined by the inequality in polar coordinates r < f(og). Some neighbourhood of any boundary point z e ~B in ~B is pinched (3) By a suitable modification of the example of Whitney [7l (see also Besicovith-Schoenberg [I]) one can obtain for 0 > 0 a real C1 function f on 12 = {(xl,x2)~ I1210 < xl < 1, 0 ~ x 2 < 1} such that the set of critical values

f(K) = f({x

l(df)=

o})

is the interval I = {t I 0 < t < 1} but for which f is a function rather close to being C 2 in the following sense. There is a closed set J (in the examples it is an arc) and for x ~ J, y ~ 12 one has, writing r = ~/(y--x) 2 , A f = f ( y ) - f ( x ) , that Af is bounded (6 > 0) r2(lnr) 2+~ and f is C ~ outside J. PROBLEM. Does there exist an example 0 = 0?

1964]

DOUBLE NORMALS OF CONVEX BODIES

79

between a unit ball and the tangent plane, mutually tangent at z, as we observed in section 3. Let n(to) be the unit outside normal vector at the point (r, t o ) = ()(co), to) of ~3B. From the assumptions about B it follows geometrically that In(to1) -- n(to2)[

Itol-to21 is bounded. The function n : S 2 ~ $2 which assigns to to the value n(co) is then totally differentiable almost everywhere by a theorem of Rademacher (see Saks [-9]). By a theorem of Federer (see Whitney [-8]) for any e > 0 there exists then a " b i g " closed set Q c S 2 such that S 2 - Q has measure < e, and such that n(to) is a Cl-function on Q. Geometrically one finds: dl-f(co).to] = f ( t o ) . ( n d t o ) t o + o r t h o g o n a l

component. The left hand side is:

f dco + (df)to The inner product with to yields, with to2 = 1 and dto 2 = 2todto = 0 , d f =f(co)(ndto)

The coefficient f(to), n(co) of dto is C 1 for to ~ Q. Hence f is C z for to e Q c S 2, and it can be extended to a C2-function g: S 2 ~ S 2 with g(to) = f ( t o ) for to ~ Q. By the theorem of A. P. Morse-Sard [6, 10] the critical values of g form a set of measure zero. The critical points of f in Q therefore give a contribution zero to the measure of the set of critical values o f f . On the other hand the critical points o f f outside Q have (Hausdorff-)measure < 5. Hence they can be covered by circular small discs with critical points as centres and with the sum of the areas smaller than 2e. Any such disc of small radius 6 in dB is again pinched between a tangent plane and a ball with radius one (section 3), and it contributes at most 2 ( 1 - c o s 6 ) < 262 to the set of values of widths of the body. The sum of these contributions is then smaller than 2/re times the sum of the areas of the discs m, hence < e. This being true for any 5, the theorem follows. REFERENCES 1. A. S. Besicovitch and I. J. Schoenberg, On J6rdan arcs and Lipschitz classes of functions defined on them, Acta Math. 106 (1961), 114-136. 2. T. Bonnesen and W. Fenchel, Theorie der konvexen K6rper, Berlin 1933. 3. W. Bos, Kritische Schnen auf Riemannschen EIementarraumstiicken, Math. Ann. lSl (1963), 434--451. 4. N. H. Kuiper, A continuous function with two criticalpoints, Bull. Amer. Math. Soc. 67 (1961), 281-285.

80

N . H . KUIPER

5. L. Lyusternik and L. Schnirelmann, Mdthodes topologiques darts les probldmes variationels, Paris, 1934. 6. A. P. Morse, The behaviour of a function on its criticalset, Ann. Math. 40 (1939), 62-70. 7. H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514-517. 8. H. Whitney, On totally differentiable and smooth functions, Pacific J. Math. I (1951), 143-159. 9. S. Saks, Theory o f the Integral, New York, 1937, p. 311. 10. A. Sard, Images ofcritlcalsets, Ann. Math. 68 (1953), 247-259. UNIVERSITY OF AMSTERDAM

OPTIMAL

SELECTION BASED ON RELATIVE (the "Secretary Problem")

RANK*

BY

Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT

n rankable persons appear sequentially in random order. At the ith stage we observe the relative ranks of the first i persons to appear, and must either select the itb person, in which case the process stops, or pass on to the next stage. For that stopping rule which minimizes the expectation of the absolute rank of the person selected, it is shown that as n --~ oO this tends to the value j=x - - 7

~ 3.8695.

1. Introduction. n girls apply for a certain position. If we could observe them all we could rank them absolutely with no ties, from best (rank 1) to worst (rank n). However, the girls present themselves one by one, in random order, and Iwhen the ith girl appears we can observe only her rank relative to her i - 1 predecessors, that is, 1 + the number of her predecessors who are better than she. We may either select the ith girl to appear, in which case the process ends, or reject her and go on to the (i + 1)th girl; in the latter case the ith girl cannot be recalled. We must select one of the n girls. Let X denote the absolute rank of the girl selected. The values o f X are 1,..., n, with probabilities determined by our selection strategy. What selection strategy (i.e. stopping rule) will minimize the expectation E X = expected absolute rank of the girl selected? To formulate the problem mathematically, let xl, ...,x, denote a random permutation of the integers 1,..., n, all n ! permutations being equally likely. The integer 1 corresponds to the best girl, ..., n to the worst. For any i = 1,-.., n let y~=l + number of terms xx, " ' , x ~ - i which are < x~ (yi=relative rank of ith girl to appear). It is easy to see that the random variables Yx, "", Y, are independent, with the distribution 1 (1) P(Yf = J ) = 7 (j ,= 1, ... ,i), and that (2) P(x, = k[ Yl =Jl, "",Yt-1 =J,-1,Y, = J ) = P(x, = =(k-1

k I y,

=j) n

Received August 22, 1964 *Research supported by Ott~ce of Naval Research and Aerospace Research Laboratories. Reproduction in whole or in part is permitted for any purpose of the United States Government. 81

Y.S. CHOW et al.

82

[June

so that

(3)

e(xi [Yi = J ) = k~=t kP(xi = kl Yi = J ) = ni___++___1_i.j.

For any stopping rule z the expected absolute rank of the girl selected is therefore

EX=E(

n+l ) . We wish to minimize this value by optimal choice of T. ---~-~y,

To find an optimal z by the usual method of backward induction we define for i = O, 1, ..., n - 1, c, = c~(n) = minimal possible expected absolute rank of girl selected if we must confine ourselves to stopping rules ~ such that v > i + 1. We are trying to find the value Co. Now

(.+

c'~-a=E

(4)

) =_1

- n - ~ l y"

n

j= a

2

'

and for i = n - 1, n - 2,..., 1,

(5)

ci- a = E

~Yi,

min

ci

= -7- ~ min

Z--,---i-J,c~

j=a

o

These equations allow us to compute successively the values c,_ a, c,_ 2 , ' " , ca, Co and contain the implicit definition of an optimal stopping rule. Equation (5) can be rewritten more simply if we denote by Ix] the greatest integer < x and set

si =

(6)

i + l ci ]J -ff--~'~

(i=n--l,...,1);

then (5) becomes (7)

I [n+l

ci-1 = -T //---+-1-(1 + 2 + ... + si) + (i - s~)ci si(si+l)

1 {n+l = -:t

i+1

2

+ (i - s3ei

}

}

.

Defining s, = n, an optimal stopping rule is, stop with the first i > 1 such that Yi < si; the expected absolute rank of the girl selected using this rule is Co. We observe from (4) and (5) that n+l (8) C0 < ¢ 1 < == "'" =0 2(i + 1)

T'(x) =

so T(x) is increasing for x < i + ½, and by (5), (8)

tl-1 ~ T(h).

We now prove the first basic inequality. LEMMA 1. (9)

2n n--i+3

ti

(i = 0,..., n - 1).

Proof. (9) is true for i~, n - 1 since by (2) t.-1 -

n 2n 2 -4"

Assume (9) holds for some 1 < i ~ n - 1; we shall prove that it holds for i - 1. We know by (1.8) and (1) that i i h - I =h--'-+-i"c~-1 _a(t /

= h - [td),

and this is true if t~ > 1, since t i < i +2 1....by (10), and if t~ < 1, since then h = ~. We now establish the second basic inequality. LEMMA 2 . 3(i + 1) tt ~_ 2(n -- i + 2)

(16)

(i = 0,..., n - 1).

Proof. (16) is true for i = n - 1; suppose it is true for some 1 < i < n - 1. Define

r(x)=x

1

2(i+1)

'

i+1 which is increasing for x < i + 1. Since by (10) h < ~ ,

ti-1 >

i

=

i+1

T(ti)>

i__~T(

= i+1

3(i+1)

we have by (14), )

2(n-i+2)

3i(4n - 4i + 5) > 3i 8(n-i+2)2 =2(n-i+3)" which is equivalent to i < n - 1. This proves the lemma. We have seen (1.9) that for any positive integer k, if n > 2k then sn-t ~ k. We now define for each k = 1,2, ..- and each n > 2k,

(17)

ik

----

smallest integer j >_ 1 such that sj > k.

We note that s ~ - I = 0 and hence from (1.7), (18)

Co = cl . . . . .

c , _ I.

86

Y.S.

CHOW

et

[June

al.

COROLLARY 2. (19)

lim i ! > 1 ..... n

Proof.

=8

•

If ix > In/2] then ix > In/2] + 1 > n/2, ( q / n ) > ½. If i x < In/2] then

by (13), l < si~ < ti: -- i~--+ l

=

i 1 +1

ia + l

n + l Cil = < n + l C [ " / 2 ] < - - n. 8+.l

=

n+l (We remark that i 1 > 1 for n > 2, since if i x = 1 then sl = 1 and Co = --~---, which only holds for n =< 2.) C O R O L L A R Y 3.

On

every set

(20)

{ ~ < - - = n i< f l ; 0 < ~ < f l < l } ,

(21)

lim (ti - t _ 1) = 0

uniformly.

Proof. From (14) and (9), 0 =< t , - q_~ =< t~

ti

i i + 1 fi

< (1 + ti) 2 < ( = 2(i + 1) =

1

}

ti

2(i+1)

-- i + 7

1 +---

l+]
12k, ik__> 1 _ 2

(22)

'

n

i__L < 1 _

(23)

n =

1

-~/~

Proof. By (9), Sik > k ~ tik > k => ~ =

=

2n

>- k =>

n -- ik --

tk> 1 n

which proves (22). (23) holds if ik < [ 2 ] , for then

i__Lk< __1 < 1 n = 2 =

1 2k'

1 • --

2 - -k'

---~ O.

1964l

87

OPTIMAL SELECTION BASED ON RELATIVE RANK

a n d i f ik > [ 2 ] then by (16), slk_x k + 1 denote the number of ( k - 1)-dimensional faces of P. With each such face F~, 1 2, then Nt e E k, its direction is that of the outer normal to Fl and its length ]N~] equals the (k-1)-dimensional content of F~. This definition associates a system ~f'(P) = {N~I 1 < i 1, f(P) > 2d. The vectors in ./if(P) span E a, i.e. the origin 0 belongs to the interior of the convex hull of the points {N, I 1 < i < f(P)}. By a Carathdodory type theorem on the interior points of the convex hull of a set, (see e.g. [6], Theorem 3.2) there exists a subset I of {1, 2,. . ., f (P)} which contains at most 2d integers and is such that 0 is in the interior of the convex hull of {N~[i ~ I}. Therefore, for suitable ~i > 0, the system ~ = {aiNi[ i~I) is fully equilibrated in E d. Obviously we may assume that the a, are such that maxi e lOgI 1. Let uff2 = { M j [ j ~ J } be the system obtained from =

{ ( 1 - a i ) N , ] i ~ I } U {N, liq~I } by the omission of zero-vectors. Since .#'(P) and ~4rl are fully equilibrated, ¢ff2 is equilibrated. Let Px and P2 be polytopes such that .A/'(P1)= .A/"x , ~g'(P2)= "#'2. If ¢ff2 is fully equilibrated, that is if P2 is d-dimensional, then the proof is completed by induction since f(P2): k}.

(7) For each sequence {x.} in E, the member 0 of T belongs to cl [O___l (conv{xi, "",x.} l conv(x.+ l, ""}) ] •

(8) For each sequence {xn} in E, the member 0 of T belongs to cl[d=l(lin{xl,...,x.}-conv{x.+l, ""})] •

104

R.C. JAMES

[June

(9) There does not exist a positive number O, a sequence {zn} in E, and an equicontinuous sequence (g,} of linear functionals such that

g~(Zk) > 0 if n k.

Proof. Clearly (1) ::> (2), since any Tl-space is countably compact if it is compact; (2) :~ (3) follows from the fact if x is an accumulation point of {x~}, then the inequality in (3) is satisfied for all f. To show that (3) ~ (4), let {Kn} be a nested sequence of closed convex sets and E 0 K~ be nonempty for each n. Choose x, from E N Kn for each n. If x in E satisfies the inequality in (3) for all f, then x ~ 0 K~. For if there is an n such that x ~ K,, then there is a continuous linear functional • with ~ ( x ) > sup {~(y): y ~ K,}.This contradicts (3), since then limtI)(x.) < ~(x). We shall complete the proof of Theorem i by showing that: (4) ~ (5) :,- (6) =~ (9),

(4) ~ (7) =~ (8) =~ (9),

(9) =~(1).

To show that (4) =~ (5), let {x.} be a sequence in E, let {fn} be an equicontinuous sequence of linear functionals for which all the limits in (5) exist, and use (4) to obtain an x that belongs to cll-conv{xn+l,-..}] for all n. Then limkf.(Xk)=f.(x) for all n, and therefore

lira lira f,,(Xk) = limf,(x). n

k

n

If L = limk lim~fn(Xk) and for a positive number 8 we choose K so that I L - limf~(x~) I < e

if k > K,

n

then I L - lim.f.(x) I _-<e. Therefore L = limJ,(x). The implication (5) =~ (6) is easy. For if {x,} is a sequence in E and (f.} is an equicontinuous sequence of linear functionals, then is bounded and there is a subsequence {(x.',f')} of {(x~,f,)} for which all the limits in (5) exist. Then the equality in (5) for {x',} and {/~} implies the inequality in (6). To show that (6) => (9), we suppose (9) is false and let 0, {z,}, and {g,} be as described in (9). Then

lYn(xk)l

inf {g~(zk): n < k} > 0 > 0 = sup {g.(Zk): n > k}, which contradicts (6). The implication ( 4 ) ~ (7) has a direct proof: If {x,} is a sequence in T, then using (4) there is an x thatis a member of cl[conv {x~+l,'"}] for all n. Then for an arbitrary neighborhood W of 0, we choose a neighborhood U of 0 such that U - U c W. For some n, there is an r in conv {xx, ...,x,} with x - r in U. For this (and any other) n, there is an s in conv {x,,+ 1,'"} with x - s in U. Then W contains r - s and (7) is verified. The implication (7) ~ (8) is formal. To show that (8) :~ (9), we suppose (9) is false and let 0, {z,}, and {g,,} be as described in (9). Let W be a

1964]

WEAK COMPACTNESS AND REFLEXIVITY

105

neighborhood of 0 chosen so that [gn(x) [ < 0 for all n if x ~. W. Then for any numbers {ai} and any nonnegative numbers {cq} with ~n+l ~, = 1, we have

Therefore W N [ U ~ = , (lin {xl,'", x,} - cony {x,+ 1,'"})] is empty. It remains to show that (9) implies (1). First we use the fact that T can be represented as a subspace of a product IIB~ of Banach spaces [12, pp. 46-47]. The weak topology of IIB~ is the same as the product topology when each B~ is given its weak topology. Since T is complete, T is strongly closed in IIB~. Since T also is convex, T is w-closed in IIBa. The canonical projection E~ of E into B~ is bounded in B~. If wcl (E~), the weak closure of E~ as a subset of B~, were w-compact for each 2, it would follow from theTychonoff theorem that the product II[wcl(E~)] is compact in the product topology when each B~ is given its weak topology. This would imply that E is w-compact, since E being w-closed in this product follows from E being w-closed in T and T being w-closed in IIBa. Thus at least one wcl(E~) is not w-compact in the corresponding B~. We can identify this Ba with a subset of a space re(A) of bounded functions by letting x be {f~(x)}, where for a suitable index set A the set {f,: 0c~ A} is the set of all linear functionals on B~ with IIf, II < 1. Since Ea is bounded, wcl (Ea) is bounded and is contained in a subset of re(A) that is a product of compact intervals and therefore is compact in the product topology. On B~, the product topology of re(A) is the weak topology of B~. Thus wcl(Ea) is not closed in re(A) with the product topology, since wcl(E~) is not w-compact. Let w be a member of re(A) that does not belong to wcl(E;.), but belongs to the closure of wcl (Ex) using the product topology. Then w also belongs to the closure of E~ using the product topology. But w is not in B~, since wcl (Ea) is w-closed in B~. Let

A = d(w, B~) in the norm (sup) topology for m(A), and choose 0 with 0 < 0 < A. We shall show that it is possible to choose a sequence {(x~,f~,)} inductively so that each f~, is one of the members of {f~: ~te A} and, with the ~ component of w in m(A) denoted by w~: (i) xn ~ E~,

(iii) f~,,,(Xk) > 0 if n < k,

(ii) f~°(Xk) = 0 if n > k,

(iv) w~, > 0.

Since II w II > 0 and w is in the product-closure of E;., it follows that w~ is a component of w iff - w~ is a component, that there is a w~, with w~, > 0, and there is an xl in Ex with f,,(xl) > 0. Now suppose that all (Xk,JJ have been chosen for k < p in such a way that (i)-(iv) are satisfied for k and n less than p. We shall choose % so that w,, > 0 and f~,(xk) = 0 if k < p. To do this, we note first that we can define continuous linear funetionals x~ (k < p) and W on B* by letting

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R.C. JAMES

x[(f) =f(x~) iff~B~,

[June

W(kL) = kw~ if ~6A.

Then it follows from +

,, = sup

aix~(f~) + W(f~) : ~

1

= sup

f~

aix k

+ w~ : ~ e A

=

aix k +

W

~

A,

that for any number @ with 0 < @ < A the Helly's condition

°H Z

+ 0 if n < p and it follows from w being in the product-closure of Ex in re(A) that there is an xp in re(A) with f~.(x~,) > 0 if n < p. Then (i)-(iv) are satisfied for k < p and n < p. Now that we have the sequence {(x,,f~.)}, let g, be defined for each n by gn(X) =f~.(Xx), where xx is the component of x in Bx. Also, for each x, choose z, in Ex for which x, is the component of z, in B~. Then g,(Zk) > 0 if n < k, and g,,(z k) = 0 if n > k. The sequence {g,} is equicontinuous, since for any e > 0 we have I g,,(x) ] < e if the component of x in Ba has norm less than e. There are other sequences of implications among (1)-(9) of Theorem 1 that can be proved easily. In particular, it would be easy to shorten the sequences used and show that (1) ~ (2) :~ (3) => (4) ~(9) =>(1). Also, Theorem 2 could have been combined with Theorem 1. This was not done because it seemed best not to interrupt the chain of implications used in Theorem 1 and because Theorem 1 is long enough as it is. The equivalence of (1) and (10) is proved in [18] for E bounded, closed and convex. Condition (12) is known [11, Theorem 6, p. 139]. Condition (14) was studied first for Banach spaces [14, Theorem 24, p. 581], but its equivalence with (1) for locally convex linear topological spaces is now well known [12, Theorem 17.12, p. 159]. Condition (16) is an interesting variation of the definition of weak sequential completeness in that limg(x,) is required to exist only for one g. Condition (17) is known for T a Banach space and K an arbitrary w-closed subset of T (see [10, Theorem 1]). THEOREM 2. Let T be a complete locally convex linear topological space and le E be a bounded w-closed subset of T. Then the following are equivalent and each is equivalent to each of(1)-(9) of Theorem 1.

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(10) Each w-continuous functional on E is bounded. (11) Each bounded w-continuous functional on E attains its supremum on E. (12) Each continuous linear functional attains its supremum on E. (13) The closure of cir (E) is w-compact. (14) The closure of cony (E) is w-compact. (15) I f {xn) is a sequence in E and limg(xn) exists for a particular w-continuous functional g, then there is an x in E with lira g(xn) = g(x). (16) I f {x~} is a sequence in g and limg(x~) exists for a particular continuous linear functional g, then there is an x in E with limg(x~)= g(x). (17) I f K is a closed convex subset o f T and E and K are disjoint, then 0 is not a member of c l [ E - K]. Proof. If ~r is an unbounded w-continuous functional on E, then the set of inverse images of the open intervals ( - n, n) is a w-open cover of E that cannot be reduced to a finite cover. Thus (1) of Theorem 1 implies (10). The implication (10) =>(11) is trivial, since if rc is bounded and w-continuous and does not attain its supremum on E, then 1 ~*(x) = sup {~(t): t ~ E} - ~(x)

defines a w-continuous functional re* that is not bounded on E. The implication (11) :~ (12) is formal, since a continuous linear functional is w-continuous. If each continuous linear functional attains its supremum on E, then this also is true for cir (E). This follows from the fact that the supremum o f f on cir (E) is equal to the supremum of If[ on E, which follows from f ( E a , x , ) < supJf(x,) I if E ] a , ] < 1.

Thus to prove (12) * (13), we could show that cir(E) is w-compact if each continuous linear functional attains its supremum on cir(E). The proof of this is difficult and known and will not be given here (see [11, Theorem 6, p. 139]). The implications (13) * (14) * (1) follow from the fact that a closed subset of a compact set is compact. Now the proof of (10) through (14) is complete. To show that (1) ~ (15), we assume E is compact and choose an arbitrary sequence {x,,) in E and suppose that g is a w-continuous functional for which lira g(x~) exists. Let this limit be L, and for each positive integer n let U,, =

/

x:lL-g(x)]

')

> n

"

Then there is an x in E with g(x) = L, since otherwise the collection of all such sets U~ would be a w-open cover of E that cannot be reduced to a finite cover. The

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implication (15) ~ (16) is purely formal and it is clear that (16) :~ (12). Thus (15) and (16) are proved. We shall show now that (1) =~ (17). For each continuous linear functional f and number • with sup {f(y): y e K} < ~, let W~ be the w-open set {x: f ( x ) > ~}. For any x in E, there is a continuous linear functional f with sup {J(y): y e K} < f(x). Therefore the set of all W~ is a w-open cover of E. If E is w-compact, this can be reduced to a finite cover. Choose e > 0 so that tI) - sup {f(y): y ~ K} > for each (f, ~) used in defining the finite cover and let U be a neighborhood of 0 such that for each s u c h f we have < ~ if x ~ u. Then it is impossible to have x - y ~ U with x in E and y in K, since we would then have x e W~ for some ( f , ~ ) and y) l < but also f(x) > • > f ( y ) + ~. Thus 0 is not a member of cl [E - K] and (1) ~ (17). To complete the proof of (17), we show that (17) ~ (12). This is easy, since if f is a continuous linear functional that does not attain its sup on E and this sup is m, then we can let K be {y: f ( y ) = m} and it is easy to show that 0 is a member of cl [E - K].

If(x) l

IS(x

-

2. Convex bounded closed subsets of complete locally convex linear topological spaces. Since a convex set is closed iff it is w-closed [12, Theorem 17.1, p. 154] we require that E be closed and convex rather than w-closed and convex. Condition (19) of Theorem 3 is a generalization of a criterion given by Pt~ik [19] for reflexivity of a Banach space, namely, that a Banach space is reflexive iff for each biorthogonal bounded sequence {(x~,fn)} the sequence {x I + - . . + xn} is unbounded. If (20) is modified by replacing O H,, by E O ( O Hn), then the equivalence of (1) and (20) becomes another theorem of Pt~k [18]. Clearly the modified (20) can be sandwiched between (4) and (20). Condition (22) with E the unit ball has been used as a characterization for reflexivity of a Banach space [16, Theorem 2, p. 1250]. Condition (23) is a strengthened form of (9), valid for closed convex sets. THEOREM 3. Let T be a complete locally convex linear topological space and let E be a convex bounded closed subset of T. Then the following are equivalent and each is equivalent to each of (1)-(17) of Theorems 1 and 2. (18) I f {(x~,f,)} is a biorthogonal sequence for which some subsequence of {f~} is equicontinuous, then there is at least one value of n for which xl + "" + x~ is not in E. (19) I f {(xn,f~)} is a biorthogonal sequence for which {x~} is bounded and {f,,} is equicontinuous, then there is at least one value of n for which xl + ' " + Xn is not in E. (20) I f {Hn} is a sequence of closed hyperplanes and E N H1 fi "" fi Hn is nonempty for each n, then N Hn is nonempty. (21) For each sequence {x~} in E, the member 0 of T belongs to

1964]

WEAK COMPACTNESS AND REFLEXIVITY cl

U(lin{x,,...,x,}-

ttat {x,+ l, ... })

109

.

n=l

(22) Each affine continuous map of a nonempty, closed convex subset of E into itself has a fixed point. (23) There does not exist a positive number O, a sequence {zn} in E, and an equicontinuous sequence {gn} of linear functionals such that

g,(zk) = 0 if n < k,

g,(Zk) = 0 if n > k.

Proof. Clearly (18):~ (19). Let us prove that (5)=~ (18). Suppose (18) is false and {(x,,f,)} is as described in (18) with {fp.} an equicontinuous subsequence of {f,}, but that xl + . . . + x, is in E for all n. Then (5) is false, since lim limfv.(x I + .-. + n

Xk) =

1 ¢ 0 = lim lim fp.(Xl + ... + xk).

k

k

n

N o w note that (20) is implied formally by (4) of Theorem 1, and (21) is implied by (8). Also, (22) is implied by (1). To see this, we use the fact that a continuous map o f a convex compact subset of a locally convex linear topological space into itself has a fixed point [2, Theorem 1, p. 82]. Since T is locally convex with the weak topology, to use (1) we need only know that a continuous affine map rc of a closed convex set K into itself is w-continuous. This can be shown easily, since if x is in K and f is a continuous linear functional, then the inverse image under rc o f {t: f(t) > f [Tt(x)] + e} is a closed convex subset o f K that can be separated strongly from x by a hyperplane. We shall show next that each of(19), (20), (21) and (22) implies (23), and that (23) implies (9) of Theorem 1. Suppose first that (23) is false and there is a positive number 0, a sequence {z,} in E, and an equicontinuous sequence {g,,} of linear functionals with gn(Zk)-----0 if n < k and gn(Zk)----0 if n > k. Let x 1 = z 1 and x~ = z n - z ~ _ 1 if n > 1. Then the sequence {x,,} is bounded and xl + "" + x:, equals z n and therefore belongs to E for all n. Also, g~(x~) = 0 and g,,(Xk) = 0 ff n v~ k, so the sequence {x,, g~/O} is biorthogonal. Thus (19) is false. N o w let go(x) be defined as lim g,,(x) for all x in T for which this limit exists and then extended so as to be continuous on all of T [12, Theorem 14.1 (iii), p. 118]. Let Ho be the null set of go and for each n > 0 let H~ be the set of all x in T with g,,(x) = O. Then z~ belongs to Ho tq H1 N ... A H~ for all n, but if x belongs to all H,,, then gn(x) = 0 for all n > 0 and therefore go(x) = 0 and x ~ Ho. Thus (20) is false. N o w suppose that W is a neighborhood of zero such that Ig,(x) I < 0 for all n if x ~ W. Also suppose that u - v is in W and

u=

~ aizi, 1

v=

~, bizi, n+l

~. b i = l . n+l

Then g~+l(U - v) = gn+x( - v) = - 0 and u - v ¢ W, so we conclude that (21) is

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R.C. JAMES

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false. To contradict (22), we first show that if x is in cl[conv {z.}] then x = ~ o ~ . z . with each ~n nonnegative and ~ . = 1. For each x in cl[conv{z.}], we define ~. for each n to be [g,,(x) - g.+l(x)]/O. Since gl(w) = 0 if w = ~ / ~ . z . with Y,/~. = 1, and all of [gl(x) and I~. - / ~ . [ for n > 0 can be made small by a suitable choice of w with each/~,, nonnegative, it follows that each ~. is nonnegative, ~ . < 1 and Y.~a,z, is convergent, and 0 = g l ( x ) = 0- ~ o . Thus ~ . = 1 and it follows that x = ~ . z . . Now let

g~(w)]

O~nZn 1

~

~ O~nZn+1. 1

Clearly rc has no fixed points. To show that ~ is continuous on cl [conv {z.}], choose a particular sequence {~,,} with each ~. nonnegative and ~ = 1. For an arbitrary neighborhood W of zero, choose a positive number 6 and a circled neighborhood Wo of zero such that Wo + Wo c W,

a " cl [conv {z.)] = Wo if I a l < fi"

By approximating a.'s in a finite set whose sum is nearly one, we can see that there is a neighborhood U of zero such that if ~fl. = 1 and each ft. is nonnegative, then El a . - fl,,I < fi if ~ ( a . - fln)Z.~ U. Then if ~-'(an -- fl.)Z.~ U, we can write this sum as the sum of those terms with positive coefficients plus the sum of those terms with negative coefficients and obtain

E ( ~ . - / ~ . ) z . + l ~ Wo + W0 = W. We must now show that (23) =~ (9). To do this, we shall assume that E is convex and (9) is false and then show that if {z.} is a sequence in E and {g.} is an equicontinuous sequence of linear functionals with g.(zk) > 0 if n k, then there is a sequence {u.} in E and a sequence {h.} such that h.(uk)=½0 if n < k, hn(u,) 0"if n > k, and each h. is equal to ~.gp for some p and some positive number ~. < 1. This can be done inductively as follows. Let li_m gx(zk) = 0'. If gl(Zk)= 0' for all k, let ul = zl and h i = (O/20')gl. If gl(zp)~ 0', choose a subsequence {zp~} of {zk} with Pk > P and [ 0 ' - gi(zp~)[ small enough for each k that there is a number 0" near 0' and between 0' and gl(zp) such that if z~ is chosen for each k so that =

z~ = aZp + ( 1 - a)Zp~ with 0 < a < 1 and gl(z~) = 0", then a is small enough that

gp.(z~) > to if n 1) and {gkl: k > 1) in exactly the same way to define u2 and h2, except that in the preceding inequality we replace 3/4 by 5/8. Continuing in this way, we get the desired sequences (u,) and {h,}. Several separation criteria are given in Theorems 4 and 5. These are closely related to (I7) of Theorem 2. A condition similar to (25) is known to be a characterization of w-compactness for closed convex subsets of a Banach space (see [10, Theorem 2]). Condition (26) is closely related to a theorem that follows from results of Tukey [21] and Klee [13, p. 881]: A Banach space is reflexive iff each pair of disjoint bounded closed convex subsets can be separated by a hyperplane. THEOREM 4. Let T be a complete locally convex linear topological space and let E be a convex bounded closed subset of T. Then the following are equivalent and each is equivalent to each of(1)-(23) of the preceding theorems. (24) I f closed convex subsets X and Y of E are disjoint, then 0 is not a member of cl [X - e]. (25) I f closed convex subsets X and Y of E are disjoint, then there is a continuous linear functional f such that sup {f(x): x ~ X } < inf{f(y): y ~ Y}. Proofi Since a closed convex set is w-closed, it follows from (1) that X is w-compact. Then (24) follows from the equivalence of (1)and (17)when E is replaced by X. We shall show that (24)=~ (23). Suppose that (23) is false and therefore for some positive number 0 there is a sequence {z.} in E and an equicontinuous sequence {g.} of linear functionals such that g.(zk) = 0 if n < k and g.(zk) = 0 if n > k. As for the set { ]~a.z.} discussed in the proof of the contradiction of (22) in the proof of Theorem 3, the following convex subsets of E are closed: X =

x

a.

~'z2.+

1 + ~'z2. n+2 Y =

, 1 an -n-T-2 Z2n--I + ~ ' ~

z2,

'

where each ~, is nonnegative and ] ~ . = 1. Then X fl Y is empty, but 0 is a member of el [X - Y]. The equivalence of (24) and (25) follows from the theorem on strong separation stated in the introduction. THEOREM 5. Let T be a complete locally convex linear topological space and let E be a convex bounded closed subset of T that contains O. Then the following is equivalent to each of (1)-(25) of the preceding theorems. (26) I f closed convex subsets X and Y of E are disjoint, then there is a continuous linear functional f and a nonzero number • such that f ( x ) < ¢p if x e X and f (y) > ¢p if y ~ Y.

112

R.C. JAMES

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Proof. Clearly (25)=~ (26). The proof that (26) implies (23) is similar to the part of the proof of Theorem 4 in which we showed that (24) =~ (23), only now we let X =

1 ~zn ~

z2"+t + ~

z2"

'

{

Y

1)}

where each e. is nonnegative and ½ < ~ e . < 1. Again X and Y are disjoint. Suppose there is a continuous linear functional f and a nonzero number 4 such thatf(x) < 4 if x e X a n d f ( y ) > 4 if y e Y. Then for all n we have f

(n+l n--~-~ z2._ t + ~-~--~ z2.

>4,

f

-~Z2n_l"~-~---'~Z2n ~f~.

This gives

f [ -nU 4+- z l

Z 2,, _

+ - f f -14 - 2 z , .

)

+

[

4

_ f ( _~__~2 z 2, _ l + _ ~2_ ~ z 2. ) ] > 2 4 "

and f[(z=._ a - Za.)/(n + 2)] > 4 for all n, so that 4 < 0. Similarly,

l(n

) O. Since 4 # O, we conclude that (26) is false if (23) is false. 3. Bounded w-closed subsets of Banaeh spaces. The equivalence of condition (27) of Theorem 6 and (1) of Theorem 1 is the classic Eberlein theorem 13]. As will be clear from the proof, if we had wished only to obtain the Eberlein theorem we could easily have proved (1) ~ (2) =~ (3) ~(27) ~ (9) ~ (1). Condition (28) is known [10, Theorem 1, p. 204] and is included here largely because of its relation to (12) and to conditions for reflexivity given in Theorem 9.

THEOREM 6. Let E be a bounded w-closed subset of a Banach space. Then the following are equivalent and each is equivalent to each of (1)-(17) of Theorems 1 and 2. (27) E is ws-compact. (28) I f S is a w-closed set and E f~ S is empty, then d(E, S) > O. Proof. First we assume (3) and let {x,} be an arbitrary sequence in E. Then there is an x in E such that limf(x,) < f ( x ) < limf(x,)

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WEAK COMPACTNESS AND REFLEXIVITY

113

for all continuous linear functionals f. Clearly x is in cl [cony {xn}]. Let {gk} be a sequence that is total over the closure of conv{xn}, and let {~} be a subsequence of {x~} for which lim, gk(~n) exists for each k. Then limg k (~,) = gk(X) for all k. n

Also, x is a weak limit of {~}, since otherwise there would be a continuous linear functional g for which lim g(~,) does not exist or does not equal g(x). Then we could choose a subsequence {t/,,} of {4 } for which limg(r/n) exists and is not g(x), and choose y for which lim f(r/~) < f(y) < lim f(r/~) for all continuous linear functionals f. Then y is in cl [conv {~/~}]. But also, lim, gk(rln) = gk(X) for all k, lim~ gk(~/,) = gk(Y), and gk(X -- y) = 0 for all i. This is impossible, since {gk} is total over cl [conv{x,}] and x ~ y follows from the two true statements: lim g(r/n) ~ g(x), and lim g(~/,) = g(y). To show that (27)=~ (9), we let {z~} be an arbitrary sequence in E and assume (27) so that some subsequence {(,} of {z,} has a weak limit w. Then w is in cl [cony {z~+l, "" }] for all n, since otherwise there would be an n and a continuous linear functional f with sup {f(Zk): k > n + l } < f ( w ) a n d thus limf((n) ~ f(w). Therefore for {z,} there can be no positive 0 and bounded sequence {gn} of linear functionals such as described in (9), since then g~(Zk)> 0 for n < k would imply g~(w) > 0 for all n, and g~(Zk) = 0 if n > k would imply g~(w) = O. Condition (28) is an easy consequence of (27), since if lim []x~ - y~ [] = 0 with x, in E and yn in S, then a weak limit of a subsequence of {x,} must belong both to E and S. Also, (28) =~ (12). For i f f is a continuous linear functional that does not attain its sup on E and m is the sup o f f on E, then the set of all x with f(x) = m is w-closed and at zero distance from E. 4. Reflexivity of Banach spaces. Theorem 7 gives many characterizations of reflexivity for Banach spaces, since each of (1)-(28) can be used as a characterization of reflexivity if E is taken to be the unit ball. The classical theorem that a Banach space is reflexive iff its unit ball is ws-compact is part of Theorem 7. The first step toward this theorem is given in Banach's book [1, Theorem 13, p.189] in which it is shown that if B is separable and the unit ball is ws-compact, then B is reflexive. In [5], it was proved that the unit ball of B is ws-compact if B is reflexive. The theorem was completed much later by Eberlein [3]. The proof given here that (31) =~ (29) was suggested by M. M. Day. A similar argument was used by Pthk to prove theorems analogous to (19) (see [19, p.321]). A weaker form of (31) is known for which g,(zk) = 0 is replaced by g~(zk) > 0 (see [9, Theorem 1, p. 206]). Condition (31) can also be stated in the following form (see [9, Corollary 1, p. 208]): It is false that for each number 0 < 1 it is possible

114

R.C. JAMES

[June

to embed B in a space of bounded functions defined on a set A in such a way that A contains the positive integers and, for each positive integer n, there is a member z,, of B with z.=(O,O,'",O,O,O,'";{tna}), where the first n components of z n all are 0 and I t~l < a for all a in A.

THEOREM 7. For a Banach space B, the following are equivalent and each is equivalent to each of (1)-(28) of the preceding theorems with E the unit ball. (29) B is reflexive. (30) It is false that for some positive number ~ there is a bounded sequence {z.} such that

d(conv {zl, ...,z,}, cony {z,,+l, ...}) >

for all n.

(31) It is false that for each number 0 < 1 there are sequences {Zn} and {g.} with I[zn[[ =< 1, I[gn[] ~ 1 , and gn(Zk) = 0 if n < k, g,(Zk) = 0 if n > k.

Proof. With E the unit sphere of B, it is clear that (30) is equivalent to (7). Therefore it is sufficient to show that (29) =:- (30) =~ (31) =~ (29). To show that (29) ~- (30), suppose that (30) is false and that {z.} is a bounded sequence with d(conv {zl, ...,z,}, conv (z,+ 1, ...}) > a for all n. Then for any x in B, there is a p such that d(x, conv{zp+l, ...}) is positive and therefore there is an f in B* with sup {f(z,): n > p} < f(x). Let F be the member of B** described in the introductory lemma, so that limf(z,,) < F ( f ) < limf(z,) if f ~ B*. Then B is not reflexive, since if there is an x with F ( f ) = f ( x ) for all f i n B*, then limf(z.) < f ( x ) < limf(z.) if f e B*, and this contradicts the fact that there is an integer p and an f i n B* with sup {f(z.): n > p} < f(x). Now suppose that (31) is false. Choose a positive number 0 < 1 and let {z.} and {g.} be as described in (31). Then (30) is contradicted by {z.} with cr any positive number less than 0, since

To prove that (31) =~ (29), we suppose that B is not reflexive and let B c denote the canonical image of B in B**. Also, for each x in B let x c denote the canonical

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WEAK COMPACTNESS AND REFLEXIVITY

115

image of x in B**. Let 0 satisfy 0 < 0 < 1 and F be a member of B** for which IIF II < i and d(F, B~) > 0. The proof will be complete if we show that it is possible to choose the sequence {(zn, g,)} inductively so that (a) I1Zn]1 < 1 and I1g, II ~ 1, (b) F(g,) = 0 for all n, (c) g~(Zk)= 0 if n > k, (d) g,(zk) = 0 if n < k. The first step is to note that since IIF It > 0, there exists g~ with IIg' [l z 1 and F(gl) = 0. Then IIel II > 0 and there exists z 1 with [[ zl II z i and gl(zl) = 0. Now suppose that (Zk,gk) has been chosen for k < p so that (a)-(d) are satisfied for n and k less than p. Then we choose gp so that IIg, IIz 1, F(g~)=O, and Z~k(gp)=gp(Zk)=O for k < p. This is possible, since the following Helly's condition is satisfied:

°1"-1

I

0 < d(F, Bc) ~ aix7 + F for all numbers {ai}, 1

where O/d(F,B c) < 1. Now we must choose Zp so that if n < p. To show this is possible, we use the fact that condition:

Izp II 1 and gn(Zp)

=

0

F t1< 1 and the Hetly's

[ ~ aiO ] = ] ~ a,F(g,) = F ( ~ a~g,) t < I{F [{ ~a~g,]]. The following theorem gives several criteria for reflexivity that are closely related and might be called "flatness criteria" for the unit ball. These are closely related to conditions (7), (8), (21) and (30). Condition (32) has long been known. It is the same as Lemma 1 of [8]. It is closely related to a necessary condition for reflexivity given by Milman and Milman [16, Corollary, p. 1252] which can be stated in the following form: If B is non-

reflexive, then for any a < 1 and any n there is a sequence {zl, ..., zn} such that a < II u II < 1 if u ~ conv {Zk} and, for all k < n, tr < d (conv

{z 1,..., Zk}, cony {zk+l,..., z,}) < 1.

To change the Milman-Milman condition and obtain the necessary and sufficient condition (32), one can replace the finite sequence {zl, ...,z,} by an infinite sequence and replace the 1 in the last inequality by 2 (which is equivalent to discarding it altogether). Condition (35) is almost equivalent to the theorem of Pdczyfiski that a Banach space is nonreflexive iff some nonreflexive subspace has a basis [17, Theorem 1, p. 372], since the inequality being satisfied for some positive number ½a is a necessary and sufficient condition for {z,} to be a basis for its closed linear span (see [1, p . l l l ] , and [7]).

116

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Condition (35) also is related closely to Corollary 2 of [23], since a sequence is a "basic sequence of type 1+" iff it satisfies the conditions in (35) with I[ u II = 1 replaced by IIu II = M for some positive number M. THEOREM 8. For a Banach space, the following are equivalent and each is equivalent to each of(I)-(31) of the preceding theorems with E the unit ball. (32) It is false that for each number tr < 1 there is a sequence {z,} such that < Ilull- tr. (33) It is false that for each number a < 1 there is a sequence {z.} such that

< II u I! ---- 1 if u e conv {z.} and, for all n,

~,...})

d(lin {zl, ..., z,}, flat {z,+

> a.

(34) It is false that for each number a < 1 there is a sequence {z.} such that a < Hu l[ < 1 if u e conv {z.} and, for all n, d(flat {zl, "", z,,}, lin {z.+ 1, "", }) > ½tr. (35) It is false that for each number < 1 there is a sequence {z,,} such that 1 i f u e c o n v { z . ) a n d , forall n < p and all numbers {a,}. n÷p ½ff ~ aizi ! • 1 t

< Ilull

Proof. Clearly (30) implies (32), (33), (34) and (35). Also, (32) =~ (33). We shall complete the proof by showing that each of (33) and (34) implies (31) and that (35) implies (29). Suppose first that (31) is false and for 0 < 1 let {z,} and {g,} be as described in (31). We contradict (33) by using the first of the equalities g " + l ( ~ a ' z ' - .+1 ~" b~z') = - 0 "

!

n+!

.+, ]~

1

n+l

with ~.+x b~ = 1 and 0 > a. Then (34) is contradicted by letting ~a~ = 1 and noting that it is impossible to have both 01 ~.+1 b, I and 011 - ~.÷1 b, I less than aif0>a. To prove that (35) * (29), we suppose B is not reflexive and denote by B c the canonical image of B in B**. For A between 0 and 1, let F be a member of B** for which IlFll < 1 and d(F,B ~) > A. The proof will consist of showing inductively that there is a linear functional • with domain B and sequences {z,} and {H,} such that II• I[ = 1, F(q~) = A, and: (i) II z. [l -_All ztl. First choose • so that I1~ 1[ = 1 and V(q3) = A. Now suppose that {z a, ...,zp} and {Hx, ..., Hp} have been chosen to satisfy (i)-(iv) when n < p, where p may be zero. Then z,+l must be chosen so that ][ z.+l 1[ < 1, ~ ( z p + l ) = A, and h(z.+ 0 = 0 if h ~ H . (if p = 0, then H p is to be the empty set). This is possible, since II F l[ < 1 and the Helly's condition A < F 11[1h + ¢ [I follows from

II

A
0 there exists g e A, g =-f on E such that

IIg IIA< IIfll ( , + We shall say that E has the extension property if for every f e A(E) there exists g ¢ A, g - - f on E and

IIg flA = II:ll E has the restricted extension property if norm preserving extension is required only f o r f e A ( E ) o f norm 1 and modulus 1. THEOREM 1.3. I f E has the restricted extension property, ¢ is affine. A pseudo-function is an element of A* with Fourier coefficients tending to zero at infinity. DEFINITION: E is an M* set if N(E) contains non-trivial pseudo-functions. E is a UM* set (uniformly M*) if E n J is either empty or M* set for every open interval J. THEOREM 1.4. I f E is UM* it has the extension property. COROLLARY. I f E is

UM*, ¢ is affine.

THEOREM 1.5. Suppose that for some ~ > 0 there exists a positive measure tt of total mass 1 with arbitrarily small support contained in E such that I--.

xl

lim sup l l~(n) I < 1 - ~. Inl--, oo

Then E has the restricted extension property COROLLARY. Under the condition of Theorem 1.5, ¢ is affine. Note that the condition of Theorem 1.5 is of a different nature than that of Theorem 1.4. A UM* set can not be too thin at any of its points while the condition of Theorem 1.5 just implies thickness somewhere.

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2. Norm-preserving extensions and affine mappings. The proofs of Theorems 1.1 and 1.3 depend essentially on the following well known Lemma:

LEMMA 2.1. Let G be a locally compact abelian group, ~ a continuous, positive positive-definite function on G, ~b(1) = 1. Then (2.1)

G, = {~; ~ G , I ~(~)1 -- a}

is a closed subgroup of G and on it @ is multiplicative. Proof. ~ = p where • denotes a positive measure on G. Now [ ~b(cr)[ = 1 if and only if a, considered as character on ~, is constant (hence = ~(a)) on the support of #. Let us return now to the situation of §l. We have ~b: E - ~ F s u c h that for f e A(F), f o dpe A(E) and

(2.2)

IlSo ~ II A,~,_-< llsll,,,r,.

S(e")= e" we obtain So I'¢'1 = ~, II '¢' II ~','~, = 1.

Taking

,~ = ,¢,, hence ,~ ~ A ( ~ and

I1'¢' II--< 1;

and since

PROPOSITION 2.2. Suppose that (a has a norm preserving extension. Then dp must be affine. Proof. By rotating E and F we may assume that ¢(1) = 1. Let ~k be a norm preserving extension of ¢, that is

¢(e") = Z(k(n)e"' satisfying (2.3)

{

~O(e~') = ¢(e i') for etteE ~l~(n)l

= 1.

Since ~b(1) = ¢(1) = 1, that is (2.4)

] ~ ( n ) = ]~1 ~(n)[,

~(n) > 0 for all n and ~b is positive definite. T~, defined by (2.1), is a closed subgroup of T, containing E, on which ~b is multiplicative; hence, for some n, ~(flt) = ei,, on T~ and the theorem is proved. As an immediate corollary to Proposition 2.2 we obtain Theorem 1.3. 3. Generalized norm preserving extensions. Let E and F be finite sets containing N linearly independent points each. Let ¢ be any mapping of E onto F. Then by Kronecker's theorem, f - - , f o ~b is an isometry of A(F)onto A(E). Clearly ¢ need not be affine and therefore has no norm preserving extension in A. In this case

124

K. DE LEEUW A N D Y. KATZNELSON

[June

it is clear, however, that ~b is generalized affine. And we prove next that such is the case in general. We shall denote by m the second dual of A (that is, the dual of yo). For a e m define

#(e") = < {e~t},a >. An element a e m is called an extension ofa f u n c t i o n f defined on some subset E of T, if

#(e l') = f ( e t') for et' e E. LEMMA 3.1. Let f e A ( E ) . Then there exists in m an extension a of f such that

[ITIIA ) --

II

II.

Proof. f defines canonically a linear functional of norm IIflIA, , on the subspace N(E) of l °°. By the Hahn-Banach theorem this functional has a norm preserving extension a e m . Now if e ~' e E, {e~n'} e N(E), and

f ( e i') = < {e'nt},f> = < {e i~t }, a> = O(e"). We turn now to the proof of Theorem 1.1. As in the proof of Proposition 2.2 we can assume ¢(1) = 1. Let tre m be a norm preserving extension of ¢, that is II II = 1, and 0 ( e " ) = ~b(e~') on E. We claim that 0(e") is positive definite on TD, To being the circle group with the discrete topology. In order to see this we can restrict tr to the closed subspace of l°° generated by the exponentials {(eint}, eUe T}, i.e., the uniformly almost periodic sequences; as such, a is a measure of mass 1 on B, the Bohr compactification of Z (and the dual of To). Since #(1) = 1, the measure corresponding to tr is positive and 0(e ~t) is positive definite. By Lemma 2.1 a(e ~t) is multiplicative on some subgroup G of T containing E; it is easy to see ([3] Th. 37) that O(e")16 can be extended to be multiplicative on T, and ¢ is generalized affine as claimed. Theorem 1.2 is an immediate consequence of Theorem 1.I and of the following: LEMMA 3.2. Let E be closed in T and a basis, Z a character of T such that Z I n is continuous. Then X is continuous. Proof. Put E , = (e~t: t = ~'~ t j, eit~ ~ E}. Since E is a basis T = U mEre, and since Em is closed for all m, we obtain, using the theorem of Baire, T =Em for m > m0. In order to show that Z is continuous we have to show that, given e > 0, there exists some interval I on T such that the variation of Z on I is smaller than 8. Put et = e/mo and E = U kj=l E~ E j being portions of E on which the variation of X is smaller than el. Clearly, for an appropriate choice of jl, "",Jmo, E' = {e~t : t = ] ~ ° t v, e it~ E E j~} contains an interval I and the variation of X on E' is smaller than mos~ = e.

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4. Sufficient conditions for the existence of norm-preserving extensions. Let E be closed in T. For 0 < e < 1 we denote by N~(E) the following set: (4.1)

N,(E)={S:SeN(E),IiSIi=I,limsupI~(n)IGo

SO that if S ~ N,(E) we have



= <S,a>

= < S, at > + < S, tr2 >

and (4.6)

<s, s > I --< I! ~, II + (1 - ~)II ~ II = Ilfll

~