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A converse to Amir-Lindenstrauss theorem in complex Banach spaces.

Ondrej F. K. Kalenda (2006)


We show that a complex Banach space is weakly Lindelöf determined if and only if the dual unit ball of any equivalent norm is weak* Valdivia compactum. We deduce that a complex Banach space X is weakly Lindelöf determined if and only if any nonseparable Banach space isomorphic to a complemented subspace of X admits a projectional resolution of the identity. These results complete the previous ones on real spaces.

A quantitative version of Krein's theorem.

M. Fabian, P. Hájek, V. Montesinos, V. Zizler (2005)

Revista Matemática Iberoamericana

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

An extension of the Krein-Smulian theorem.

Antonio S. Granero (2006)

Revista Matemática Iberoamericana

Let X be a Banach space, u ∈ X** and K, Z two subsets of X**. Denote by d(u,Z) and d(K,Z) the distances to Z from the point u and from the subset K respectively. The Krein-Smulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w*-compact subset K ⊂ X** such that d(K,X) = 0 satisfies d(cow*(K),X) = 0.We extend this result in the following way: if Z ⊂ X is a closed subspace of X and K ⊂ X** is a w*-compact subset of...

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