THE STRONG WCD PROPERTY FOR BANACH SPACES

In this paper, we introduce weakly compact version of the weakly countablv determined (WCD) property, the strong WCD (SWCD) property. A Banach space X s said to be SWCD if there s a sequence (A,) of weak. compact subsets of X** such that if K C X is weakly compact, there is an (n,) N such that h'C = 1An C X. Inthiscase, (A,)iscalleda strongly determining sequence for x. We show that SWCG SWCD and that the converse does not hold in general. In fact, x is a separable SWCD space if and only if (X, weak) is an 0-space. Using o for an example, we show how weakly compact structure theorems may be used to

the closed unit balls of x and X** respectively.
X is said to be weakly compactly generated (WCG) if there is a weakly compact K C X with the span of K dense in X [3].The WCG property has been an active topic of research for several years (e.g., ([1], [3], [8]).Similarly, a generalization of this property, the WCD property, has been investigated, pa:ticularly since WCD spaces possess many of the same properties as WCG spaces (e.g., [6], [11], [12]).x is said to be WCD if there is a sequence (A,) of weak.compact subsets of X** such that for each z e X there is an (n) c N with e f'l= A, C X [12].In this case, we say that (A,) weakly determines X.We will see that each of these properties may be expressed as a property of the family of norm compact subsets of x.Our goal here is to introduce the weakly compact version of the WCD property, the SWCD property, and to examine its relationship to the strong WCG property of Schltichtermann and Wheeler [9].
We first state some definitions and results.
X is strongly WCG (SWCG) if there is a sequence (K,,) of weakly compact subsets of X such that for each weakly compact subset H of X and each >0, there is an hen such that H C K, + B [9].As noted in [9], restricting H to norm compact sets in the above definition gives a definition of WCG that is equivalent to the one above.
x is SWCD if there is a sequence (A,) of weak.compact subsets of X** such that for each weakly compact K C X there is an (n,)c N with K C f']= 1An C X.In this case, we say (An) strongly determines X.
The following result affirms the claim that SWCD is the natural definition for the weakly compact version of WCD.PROPOSITION 1. x is WCD if and oulv if there is a sequence (..1,,) of weak.compact subsets of X** such that for each noru ('Oral)act K X, there is an (,,,,)c N with Uc N=A, cx.PROOF.() Clear.() Suppose (C,) is a sequence of weak* compact subsets of x** that veakly determines x.B** Now let (A,) be an enumeration of the finite unions of the For each , j E N let F,, ' +7 F,,, and note that each A,, is weak compact.Suppose K is a norm compact subset of X. Choose a sequence (n)C N so that E (nn)K C A,.We certainly have K C N .4,,so we need only show that M A, C X.
Le **E X**kX.For each z E K there is an ()E N such that EC,() and ** B**.Note that this lmst set h nonempy There is also a j() E N such that ** C,(.) + norm inerior, so, in fac, we may find ,...,.K such that =1 and ...) he se on the right is one of the A containing K, hence it is one of the A so we have ** 8= A, .
We have WCGWCD from [12], and the analogous result for the stronger properties from the following.PROPOSITION 2. If X is SWCG then x is SWCD.B** wih j,n N, where PROOf.Let the (a,) be enumeration of sets of the form nK + K is an SWCG generator for X.B I is well-known that WCD spaces e Lindel6f in he we opology [12].Utilizing strengthening of the Lindel6f property, we have a simil result for SWCD spaces.
A family of subsets of a opologic space T is called a stron 9 open cover of T, if open cover of T d for each compac subset K o T there is a U 6 with K C U. If every strong open cover of T has a countable strong open subcover, T is sd to be strongl Lindel6f (SL).This property w first studied in ([4], [5]) in relation to properties of the compact-open on spaces of continuous functions.PROPOSITION 3. If X is SWCD then (x, weak) is strongly Lindel6f.
PROOf.Suppose (A,) is a sequence of we.compact subsets of X'* strongly determining X, d let {U} be a strong open cover of (X, wea).or each 6 1 there is n Le q be the collection of 1 finite subsequences, , of N such that , ,A, V for some a 6 1. Infinitely my such exist since there e infinitely my (n) c with A X.
We may sume q {i}= a. Eor each 6 chse u, 6 1 such hat ,,Aj C Le K be a wetly compact subse of X.By hypothesis there is (n) c such that K = aAn X.There is also a 6 1 such that K C = An C U C V, hence, since V c X** is we.open, here is a 6 such that K C ', A, C V. Now m, .,mi o some i, so K C Vai. Since K C X, we have K C U,. herefore, (U,) is a countable strong open subcover of .B he k of identifying SWCD spaces may be reduced by heorem 1.In orde to prove this esult, we fecal he following definition.
Let T be a completely rel topological space, d let 9 be a fily of subsets of T. 9 is said to l, a l"doba" f 7" if fl ',(h open c "1 and each conpact !c there is a such that K C l'C l" If "I has a tntalle pse(lol,,e.I" is said to t, an Ro-sIace [7].A recent study of s0-space, in regard to Banach spaces i, giw'n in [10].In [7] it i, prv,,,1 that f T is an R0-space then T is Lindelaf, and, in fact, an eh'mentarv modification of this proof reveals that "I is strongly Lindel6f.
The following esult indicateb that sepatallity provides a way t "'isolate"" a w'aklv compact convex set K flora XK ing intersection of menl)er of a countal)h' fanlv of weak.compact subsets of X'" The SWCD property is the (ondition needed to isolate A flown .V'*X.V.
will be an N0-space precisely when K is isolat('d from X"K in this manner. LEMMA 1.Let X be separable.Then the're is a sequence (F,) of ,., compact subsets of such that if K is a weakly compact convex subset of X, there is an (,,,)c with K N : =, F. )Nx.
PROOf.Let the norm on x be denoted by 1. I. From [1] there is an equivalent norm.
II1-111 on x, uch thai every weakly compact convex subset of X can be written as the intersection of closed II1" II-bls.Now let (,) be a dense sequence of points in {x, lll-IIl.Suppose that K is a weakly compact convex subset of X and EX.Let zEx and a>0 be such that KCB(z,b) and B(z,b), where the ball is the closed ball with respect to II1 Ill.We clearly" may enlarge this closed bll o rdius so that r-a >0, is rational, and B(,,,.).Set =,,{{-a),lll-lll-), and find n so that II1-111 < .Then (z,p)D K.This shows that for X separable, it, is enough to consider those closed balls centered at some z and of rational radius in the previous paragraph.Let the (Fn) be an enumeration of the m, closures in A"* of this collection of closed III.Il-b.
THBOM 1.If X is separable, the following are equivalent.
() xisSWCD.() (X, wea) is n R0-spce.() There is a sequence (A,) of m, compact subsets of X'" such that if K is a weakly compact convex subset of x, then there is an (n=) c with K n A, m- PROOf.(12).Suppose (An) is sequence of m, compact subsets of X*" that strongly determines X. Choose the sequence (F,,) according to Lemma and let (U,) denote n enumeration of the members of (A,) and (F).Then let '= (P) be a sequence formed from 11 finite intersections of members of (C), nd set (P,) where P, P N X for ech E .
By the sub-base theorem in [7], it is enough to show that for ech weakly open convex set UcX nd wekly compact convex set KCU, thereis n nE such hat KCP, CU.Assume d K re given this way, where u=vnx for some m, open Vcx*'.Then thereis (nm) C such ht K= n=u Hence here is E such that n m=U.cv, but n=G, =P' for some j, so KcPcV.Thus KCPCU.
(2a).Assume h& (X, wea) h&8 & countable pseudobse, =(P).Without loss of generlity ssume that ech member of is bounded nd wekly closed in x. or ech let A c x'* and note that ech A, is compac in Let K C X be wekly compact, and choose (n) c so that E (n)K C A,. Suppose X**K.Then there is m, open set V c x*" such that K C V nd *" g '.By hypothesis, there is an nE such that KCP, CV, so KC'CV', nd hence KCA nd ,'*A,.Therefore = N= a, (al).Obvious.
It should be noted that SWCD ds not imply separability, since 11 reflexive spces are SWGD, nd separability ds not imply SWCD, because there exist separable Bnch spces, I).WILKINS C([0,1]) for instance, which are not 0-space in the weak topology [7].A simple exainple of an SWCD space that is not SWCG is given in the following. EXAMPLE.
Since o is not weakly sequentially oomph're it cannot be SWCG [9].However, (co,weal:) is an 0-space, since it has separalle dual [7].so o is SWCD.The following example demonstrates how strongly determining sequences may be produced by utilizing results al)out the structure of weakly compact sets.
From [2] we obtain the following result.
Let M C o. Then M is relatively weakly compact if and only if M is bounded and for every (rnt) c N we have .m: sup ( in, 1)--,0, M 1,

Xrnk
Let e be the collection of all finite subsequences of N. For n, N and e, set Then each A .... is bounded and w. closed, hence w, compact.The collection of all A .... is countable, so let C be an enumeration of the A ' ri t" Suppose K is a weakly compact subset of 0. Let (m)c N be the collection of all n hr such that K C C,, noting that there are indeed infinitely many such C,, by the above result.Now suppose z'*6 e\c0.Then there is a j 6/v and a (t)c/v such that z;' > 1/j for all k >_ 1.By the above result [2] again, K C A . . . .for some 6 h" and r of the form r t,t,. .,t,,yet z*" is contained in no set of this form.Thus z** i"1= 1C,,.Therefore f'l= C, c X, hence (C,) is a strongly determining sequence for 0.