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On equivalent characterizations of weakly compactly generated Banach spaces. (English) Zbl 0877.46006

Let us consider the following nice
Theorem. For a Banach space \(V\) the following assertions are equivalent:
(a) \(V\) is weakly compactly generated (w.c.g.),
(b) \(V\) is GSG and simultaneously a Vašák (i.e., weakly \(K\)-countable determined) space, and
(c) \(V\) is GSG and moreover \((V^*,w^*)\) continuously injects into \(\Sigma(\Gamma)\) for some set \(\Gamma\).
A main aim of this paper is to present a more direct proof that (b) or (c) implies (a). We shall avoid interpolation as well as gymnastics involving Gul’ko and Corson compacta. In particular, we shall no longer need a result of Gul’ko that a continuous image of a Corson compact is a Corson compact.
A central concept, we shall use in our proof will be a slight variant of a projectional generator introduced recently by J. Orihuela and M. Valdivia [Rev. Math. Univ. Complutense Madrid 2, 179-199 (1989; Zbl 0717.46009)].

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces

Citations:

Zbl 0717.46009
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References:

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