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A note on Asplund generated Banach spaces. (English) Zbl 0946.46016

A Banach space \(X\) is called Asplund generated if there exist an Asplund space \(Y\) and a bounded linear operator \(T:Y\to X\) with dense range. Let \(X\) be a Banach space with density character \(\aleph_1\). The main result of this paper asserts that \(X\) is weakly compactly generated if and only if \(X\) is Asplund generated and satisfies one of the following conditions:
(i) \(X\) is weakly Lindelöf determined;
(ii) \((B_{X^*}, w^*)\) is a Corson compact;
(iii) for some equivalent norm \(X\) admits a projectional resolution of identity (PRI) satisfying a suplementary condition;
(iv) the space \(X\) admits a PRI for any equivalent norm;
(v) for some equivalent norm \(X\) admits a PRI and \((B_{X^*}, w^*)\) has countable tightness;
(vi) \(X\) admits an equivalent Gâteaux smooth norm \(||\) and a PRI with respect to \(||\);
(vii) \(X\) admits an equivalent Gâteaux smooth norm \(||\) such that the corresponding unit ball \(B_{X^*}\) is a Valdivia compact with respect to the topology \(w^*\).
The obtained results are illustrated on the case of the space \(C[1,\omega_1]\). A good source for the notions used above and for related results is the fine book: M. Fabian, “Gâteaux differentiability of convex functions and topology. Weak Asplund spaces”, New York (1997; Zbl 0883.46011).

MSC:

46B20 Geometry and structure of normed linear spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
46B26 Nonseparable Banach spaces
46B50 Compactness in Banach (or normed) spaces

Citations:

Zbl 0883.46011
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