On some new characterizations of weakly compact sets in Banach spaces

• Autorzy: Lixin Cheng , Qingjin Cheng , Zhenghua Luo
• Języki publikacji: EN

Abstrakty

EN

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*.

Kategorie tematyczne

• 46A55: Convex sets in topological linear spaces; Choquet theory
• 46B20: Geometry and structure of normed linear spaces
• 58C20: Differentiation theory (Gateaux, Fr\'echet, etc.)
• 46B03: Isomorphic theory (including renorming) of Banach spaces
• 46G05: Derivatives
• 46B50: Compactness in Banach (or normed) spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 201
• Numer: 2
• Strony: 155-166

Daty

wydano

2010

Twórcy

autor

Lixin Cheng

• School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

autor

Qingjin Cheng

• School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

autor

Zhenghua Luo

• School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

Identyfikatory

DOI

10.4064/sm201-2-3