Vašák, L.

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Institute of Mathematics AS CR, v. v. i.

WSAA 4Lubomír VašákOn one generalization of the weakly compactly generated B-spacesIn: Zdeněk Frolík (ed.): Abstracta. 4th Winter School on Abstract Analysis.Czechoslovak Academy of Sciences, Praha, 1976. pp. 147--149.Persistent URL: http://dml.cz/dmlcz/701065Terms of use:© Institute of Mathematics of the Academy of Sciences of the Czech Republic,1976Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copy of anypart of this document must contain these Terms of use.This paper has been digitized, optimized for electronic deliveryand stamped with digital signature within the project DML-CZ:The Czech Digital Mathematics Library http://project.dml.czPOURTH WINTER SCHOOL (1976) ON ONE GENERALIZATION OF THE WEAKIX COMPACTLT GENERATED B-SPACES by L. VA££K Because reflexive B-spaces are exactly those* which are S-compact in their weak topology, the following generaliza-tions of ^-compactness are of interest: 1) analytic topological spaces ( a topological space T is called analytic iff there exists a mapping f from cu $ **J being the set of all finite ordinal numbers and also so the first infinite ordinal number into the set of all compacts in T such that for any open G in T the set -Cnre to • f (1T) € € G $ is open in usual product topology of a;, . 2) Topological spaces which arel^j- ( a topol* space T is called K^r in a topological space T iff T«T' t the topologies of* T and T coincide on T and there are * -* «© compacts A ^ in T so that T « ^C\ \j A ^ ) • It is easy to show that for any topol. space T it holds: (T is 6"-compact)*•> (T is K&* in some top. space T') «-^(T is analytic) —»> (T is LindelSff in its weak topology). Definition 1: A B-space X is called weakly analytic (wA) iff X is analytic in its weak topology. Definition 2. A B-space X is called weaker it (WK) iff X is X& in X** (second dual in its w -topology). Remark 1. It can be shown that if a B-space X is £44* in some uniform space T', then X is WK.so the definition of WK property is not too restrictive. Proposition 1: Let X be a WCG B-space- Then X is WK (with convex AAAt 's - we will call this CWK property vor space)) . This was proved independently by D. Pre iss and Tala-grand and by means of this observation they solved this pro-blem of J. Lindenstrauss: Is every B-space WCG iff it is LindelCff in its weak topology? So the implication " «=> " holds and the opposite can-not be true because WCG property is not hereditary (on clo-sed linear subspaces) and LindelSff property is. Many of basic properties of WCG B-spaces can be proved for CWK 3-spaces; Theorem 1. Let X be a CWK B-space. Then: (a) X has a projectional resolution of identity i.e. there are linear projections P_ •> w -. c* - je (^C-is the first ordinal number of cardinality dens X =- inf i card H ; H is dense in X \ ) such that: (i) II Pji = 1 for any oo * O -£ cC • U , pa6 = identity on X , (ii) ^ Pp = Pp P^ = ̂  for any <* -fr «6 .£ /3 £ 4* f (iii) dens PoC,(X)^card oc , (iv) the function P^ (x), x s l fixed, is continuous on <ca,'K>> in the usual order topology. (b) dens X = W* - dens X (= density in >.* -topology) (c) there exist a set V . and a linear continuous one-to-one mapping from X into c0 C V ) * (d) X admits an equivalent locally uniformly rotund (LUR) norm. (e) X has an equivalent Frechet differentiable norm iff X has an equivalent norm which dual norm on X* is LUR iff X has a shrinking Markusevic" basis. (f) If X* is also CWK, then X is WCG. Proofs of this theorem are mainly generalizations of tho-se for WCG B-spaces. They are in some cases more simple. Be-cause CWK property is hereditary, every subspace of WCG B-spa-ce is CWK, but while we do not know any definite way how for given subspace of WCG B-space to find this WCG space,for CWK spaces this ambiguity is overcome by Remark 1. Problems: Problem 1. All converses to the following implications: A B-space X is: (subspace of WCG B-space)«.> (CWK).—* (WK)»*(WA) = > s ^ (weakly LindelOff) ?! Problem 2. Is Theorem 1 true also for X only WK (WA, weakly LindelOfT) ? Problem 3. Let X* be a CWK B-space. Is'then a) X* with Radon-NikocV̂ m property, b) X Asplund space (i.e. strong differentiability Space) ? 

On one generalization of the weakly compactly generated B-spaces

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On one generalization of the weakly compactly generated B-spaces

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Institute of Mathematics AS CR, v. v. i.