Characterization Of Weak Compactness Research Articles

The theorem of Halmos and Savage under finite additivity

We prove an extension to the finitely additive setting of the theorem of Halmos and Savage. From this we deduce extensions of classical results of Drewnowski and of Yan as well as a new characterization of weak compactness in the space of finitely additive set functions.

AN ELEMENTARY PROOF OF JAMES’ CHARACTERIZATION OF WEAK COMPACTNESS

AbstractIn this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.

Measures of weak noncompactness in Banach spaces

Measures of weak noncompactness are formulae that quantify different characterizations of weak compactness in Banach spaces: we deal here with De Blasi's measure ω and the measure of double limits γ inspired by Grothendieck's characterization of weak compactness. Moreover for bounded sets H of a Banach space E we consider the worst distance k ( H ) of the weak ∗-closure in the bidual H ¯ of H to E and the worst distance ck ( H ) of the sets of weak ∗-cluster points in the bidual of sequences in H to E. We prove the inequalities ck ( H ) ⩽ ( I ) k ( H ) ⩽ γ ( H ) ⩽ ( II ) 2 ck ( H ) ⩽ 2 k ( H ) ⩽ 2 ω ( H ) which say that ck, k and γ are equivalent. If E has Corson property C then (I) is always an equality but in general constant 2 in (II) is needed: we indeed provide an example for which k ( H ) = 2 ck ( H ) . We obtain quantitative counterparts to Eberlein–Smulyan's and Gantmacher's theorems using γ. Since it is known that Gantmacher's theorem cannot be quantified using ω we therefore have another proof of the fact that γ and ω are not equivalent. We also offer a quantitative version of the classical Grothendieck's characterization of weak compactness in spaces C ( K ) using γ.

Weak compactness and uniform convergence of operators in spaces of Bochner integrable functions

The classical characterization of weak compactness in the space L’(p) over a measure space (X, C, p), states that a bounded set Kc L’(p) is relatively weakly compact iff it is uniformly o-additive (or, equivalently, uniformly inegrable, or uniformly p-absolutely continuous, in case P-l(X) < 00 1. In this paper we give a new characterization of weak compactness in L’(p) which is completely different from the classical one. It is stated in terms of uniform weak convergence of certain “admissible” sequences of operators (Theorem 6). It is interesting to make a comparative analysis of these two characterizations of weak compactness.

Weak compactness and reflexivity

Several characterizations of weak-compactness are given for subsets of complete locally convex linear topological spaces and of Banach spaces. Some are new and some are generalizations of known facts.