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On some new characterizations of weakly compact sets in Banach spaces

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Cheng L., Cheng Q., Luo Z.

Studia Mathematica

2010

tom 201

nr 2

155-166

We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w*-lower semicontinuous seminorm p on the dual X* with $p ≥ sup_{C}$ such that p² is everywhere Fréchet differentiable in X*; and as a consequence, the space X is a weakly compactly generated space if and only if there exists a continuous and w*-l.s.c. Fréchet smooth (not necessarily equivalent) norm on X*.

Ograniczanie wyników

1 Studia Mathematica

1 Cheng L.

1 Cheng Q.

1 Luo Z.

1 2010

Wyniki wyszukiwania

On some new characterizations of weakly compact sets in Banach spaces