Commentationes Mathematicae Universitatis Carolinae

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1 Commentationes Mathematicae Universitatis Carolinae Jiří Reif A note on Markuševič bases in weakly compactly generated Banach spaces Commentationes Mathematicae Universitatis Carolinae, Vol. 15 (1974), No. 2, Persistent URL: Terms of use: Charles University in Prague, Faculty of Mathematics and Physics, 1974 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Commentationes Mathematieae Universitatis Carolina* 15,2 (1974) A NOTE ON MARKUgEVlC BASES IN WEAKLY COMPACTLY GBNERATBP BANACH SPACES Jif i REIF, Praha Abstract: The concept of Markulevie* bases is used to give more elementary proofs of some results on weakly compactly generated Banacrf spaces. Keir words: Weakly compactly generated Banach spaces, MarkoSeviS basis. AMS: 46B15 fief notation: By normed linear space, we shall mean a real oar, by trfu A for a set A we denote the. linear spaa of A, 3Sfi A denotes tae claaare af Aft<A * POT a locally convex space X, by X* we mean the dual of X (i^e. coatinoaaa linear functions, on X ). A Jtamretr space X is called weakly compactly generated (in siiart W ) if there exists a weakly compact set XCLX such tnajt JJfvX m X, A Mortihagonal system ix^ >.{, J* «i i*-- X x X* ( X is in general a locally convex space) is called a MarkuSevifi basis {i siiort it. -basis) if. $, (*.) -» dti fqr -S ^ * 5^ Axj, flgj *.X, and $, (#) «0 f or all -i* e 1 implies ix m 0, By /i<r (resp. 4<r* ) we mean the &(X, X ) ' (r&&&+ &(X* 9 X) } topalagy.. c 0 C V ) denotes the space of

3 real valued functions x on P suck, that for. each;... --> > 0 the set -t^jr * T $ } x C^ ) I > e? is finite, Ux II *ttf* \x (^r)l» o> is the first infinite ordinal number* For a topological space X we denote by wx the smallest cardinal number tf such that there exists a set AcJ[, cwtci A m^>. A is dense in X. We say that a locally convex space is generated by A if X m Sfi A Learn- (ill)* Let X be a WCG space, generated by a weakly compact absolutely convex set X..Denote $ tha first ordinal of cardinality mrx. Then there exists a systea -CPoc Ja># oc.6 r of iinear projections such that iu^/!*. «4, P A KcK, w CP^ X ) m OOJUL oo for each oc, V ^ X j'fgcfy- %?<* - Poc for each oc /3, ^X» *-* ^^3X for each ec- limit ordinal. Proposition. Let X be a WCG space and X c X be a weakly compact absolutely convex set generating. X.Then there exists a MarkuSeviS basis "C-<i >.i?-i e I that K i, e X for i el * of %' such-' Proof. We prove Proposition by transfinite induction on <wx.let X be separable. Then A^jvX is a separable normed linear space and there exists an JA. -basis of 4>ft X",' which is also an JA -basis of X (see e.g. C3J). For this M -basis <X^,f.i^ej we can suppose X^eX as X afc^ sorbs elements of &ft> X. Suppose now that ntrx > -Ko a^ that Proposition has been proved for all spaces 7 with *ry <: wx. Let ****»*«*% be the s 7 stenl of P r

4 jactions* from the above lemma Then Y^ * F^ X ia a WCG space generated by weakly compact absolutely convex set X-o; = P<tf&, ^u> ss G&tcLco<:<urX. Similarly 3^+4» CF^.n - - P^ > X ia generated by weakly compact absolutely convex set Xflt + 4» -j C P**,* - P^ ) X for CJC < J, and ^trj^ < <wrx for all- these oc., By the induction hypothesis there exists for each oc < $ an Jii -basis of ^t'^ugl* 7 * with- x* ** X* Uml«)'.. Now **?,?# P*A «T u {*? % C P_, -P.)}.,! is obviously an JA -basis of X and ****X-T *-*c ^X«* "^ * \c» * oc-c$ * c c X. Corollary! A Banach space is WCG if and only if there exists an M -basis 4**,, ih«i of X such that 4*iii, l ^*0? is * akly compact. The coefficient a i^i^gi f such an M -basis in a WCG space X can be found in an arbitrary weakly compact absolutely convex set X generating X Proof* The part "if" of our assertion is trivial a* X is generated by ix±\\,*i Lat X be a WCG space and X be a aet as above. By Proposition there exists an it -basis 4*4,, 4, J-i #1 of X such that A m i^^iei c X. We must only prove that the only <ur -cluster point of A is x» 0 C A is obviously discrete in tv topology) * Let tf be a cluster point of A, and # be the limit of a net <*,> J^cA c ^ Kxen * or an arbitrary -i, «I is ** 4* *4 f or» i> 0 for some ox, e A. Thus» (x>=- &m, ±(x») - 0 which implies

5 X m 0 as 4 ^J^ l is total on X Remark I. The following two results are due to Amir, Corson and Lindenstrauss (t!3, C43 ) However, they used for their proofs a measure representation theorem and the Stone- Weierstrass theorem. Corollary 2. Every WCG space is generated by a set which is in the weak topology one point compact if ication of a discrete set. This set can be found in an arbitrary weakly compact absolutely convex set X generating the space. Proof, for the generating set take the set ix±$jiei v u <0} from Corollary 1. Corollary 3. Let X and *C*i>f.i H e 1 De as in Corol ~ iary 1. Then the mapping Ts X* * c oci) defined by TC ) =s -C.X4,l.v«I * s a 1 ~ 1 -*in e ar <ur*~ *ur continuous mapping onto a dense subset of c 0 CI) - Proof. The /ur^-ur continuity follows from the theorem of Banach-Bieudonne". Remark 2. Let X be a locally convex space and 4*4,, 4 l-i-cl an ^ "~ oasis &* % sacn that K R,.^?i 8 x is relatively wr compact. Then the mapping T. X* c 0 CI), TC ) a? <i (xj^)}^ej is a 1-1 linear mapping which is >ar*- no- continuous on K m it e X*j i Cfe)l 4 \ for Two following simple examples satisfy assumptions of

6 .Remark 2. Example 1«Let Y be a normed linear space with an J/l -basis *f ^.; ; f.i?i 6 1 * F e can suppose that II 4,11j 4 for i- e I. Denote X = y* with some topology which coincides with the duality <y,y*> # Then <f«s,,#ii-s, *x is an.m -basis of X. Example 2. Let Y be as in Example 1* We can suppose that HXJ^ I # 4 for iel ( 4, can be unbounded). Let 2 cj* be a subset, Z ----C -L ^iel? such that /Cx^^gj is &CY f Z) relatively compact (for example 2 */5fif ^ej)» Denote X «7 with some topology which coincides with the duality <y,z> * Theorem 7 of T23 due to C. Bessaga, A. Pelczynski and S. Trajanski and Proposition 3,4 of 141 due to H. Corson (see Corollary 2 above) combine to give Corollary 4. Let X be an absolutely convex weakly compact subset of a Banach space X. Then the space CCK.) of all continuous (with respect to w topology on X ) real valued functions on X is homeomorphic to the space >fif Cf) where =* mr C C CK))» /wx. Proof. Denote Y m J>fi X, and ^ be the first ordinal of cardinality wy. (There is <ury& wk.) Let L be the generating set of T in X from Corollary 2. L is one point compactification of a discrete set and so CCL) is homeomorphic to ^ ( ), (Trojanski (Th.7»t2l)). As Y. is a normed linear space there exists a set Z c Y >

7 &axd> Z m owed J, Z separates points of T. Thus we have L c X, CCL) «4f/ > and <ur (CCX)) S c<ww $ and thus Ajf ( C C.K)) m COKCL by Stone-Weierstress theorem. The* refore CCJQ ia. nomeoj^ to JL CJ) lth.7,121). Heferences tl] D. AMIR and J. LINDENSTRAUSS: The structure of weakly compact sets in Banach spaces, Ann.of Math.88 (1968), [2] C. BESSAGA and M.I. KADEC: On topological classification of non-separable Banach spaces, Proc.Symp. Infinite Dim.Topology 1967,Ann.of Math.Studies,.Princeton,N.J.,69(1972),l-7. [3] W.B. JOHNSON: MarkuSevic* bases and duality theory, Trans.Amer.Math.Soc.149(1970), Matematicko~fyxik ln fakulta Karlova universita SokolovskA 83, Praha 8 Ceskoslovensko (Oblatum ) [43 J. LINDENSTRAUSS; Weakly compact sets,their topological properties and Banach spaces they generate, Proc. Symp.Infinite Dim.Topology 1967, Ann.of Math.Studies,Princeton,N.J.,69(1972)