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).


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


Zbl 0883.46011