### On locally uniformly convex and differentiable norms in certain non-separable Banach spaces

S. Troyanski (1971)

Studia Mathematica

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S. Troyanski (1971)

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Lixin Cheng, Qingjin Cheng, Zhenghua Luo (2010)

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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 $p\ge su{p}_{C}$ such that p² is everywhere Fréchet differentiable in...