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## Studia Mathematica

2010 | 201 | 2 | 155-166
Tytuł artykułu

### On some new characterizations of weakly compact sets in Banach spaces

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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*.
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Tom
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155-166
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wydano
2010
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• School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
autor
• School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
autor
• School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
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