Sequential w-right continuity and summing operators

arXiv: Optimization and Control, 2016

We provide sufficient conditions for a Banach space Y to be weakly sequentially complete. These conditions are expressed in terms of the existence of directional derivatives for cone convex mappings with values in Y .

Bulletin of the Australian Mathematical Society, 1993

Let X be a completely regular space, E a Banach space, Cb(X, E) the space of all continuous, bounded and E-valued functions defined on X, M(X, L(E, F)) the space of all L(E, F)-valued measures defined on the algebra generated by zero subsets of X. Weakly compact and β0-continuous operators defined from Cb(X, E) into a Banach space F are represented by integrals with respect to L(E, F)-valued measures. The strict Dunford-Pettis and the Dunford-Pettis properties are established on (Cb(X, E), βi), where βi denotes one of the strict topologies β0, β or β1, when E is a Schur space; the same properties are established on (Cb(X, E), β0), when E is an AM-space or an AL-space.

Glasgow Mathematical Journal, 2006

In this paper we present some results about wV (weak property V of Pełczyński) or property wV * (weak property V * of Pełczyński) in Banach spaces. We show that E has property wV if for any reflexive subspace F of E * , ⊥ F has property wV. It is shown that G has property wV if under some condition K w * (E * , F *) contains the dual of G. Moreover, it is proved that E * contains a copy of c 0 if and only if E contains a copy of 1 where E has property wV *. Finally, the identity between L(C(, E), F) and WP(C(, E), F) is investigated.

Bulletin of the Australian Mathematical Society, 1989

Let X be a completely regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremum-norm.In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

Indagationes Mathematicae (Proceedings), 1973

Acta Mathematica Hungarica, 2018

The p-Gelfand Phillips property (1 ≤ p < ∞) is studied in spaces of operators. Dunford-Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T : X → Y is weakly compact, for every Banach space Y .

Journal of Convex Analysis

We give a counterexample to a recent statement in the metric approximation theory and provide a setting where the statement holds. Let (X, •) be a real Banach space. Our notation is standard. We follow, for example, [FHHMPZ01]. In this note, if no reference to a different topology on X is made, convergence in X means •-convergence. The following concepts in the geometry of Banach spaces are more or less standard. A non-empty subset C of X is said to be approximately compact if for every x ∈ X and every sequence (c n) in C such that x − c n → dist (x, C) (such a sequence is called an approximate sequence for x in C), then (c n) has a convergent subsequence. The set C is said to be proximinal if, for every x ∈ X, the set P C (x) := {c ∈ C; x − c = dist (x, C)} is non-empty (the multivalued mapping P C : X → 2 X is called the metric projection onto C). The set C is said to be semi-Chebyshev if P C (x) contains at most one point for every x ∈ X. The set C is said to be Chebyshev if it is simultaneously proximinal and semi-Chebyshev. In this case we put P C (x) = {π C (x)} for all x ∈ X. A Banach space X is said to be locally uniformly rotund (LUR, for short) if for every x ∈ S X and every sequence (x n) in S X such that x + x n → 2, then x n → x. A Banach space X is said to be midpoint locally uniformly rotund (MLUR, for short) if for any x 0 , x n and y n in S X , n ∈ N, such that x n + y n − 2x 0 → 0, then x n − y n → 0. Every LUR Banach space is MLUR. Recall, too, that a Banach space X has property (H) (sometimes also called Kadec-Klee property) if every sequence in S X that w-converges to a point x in S X converges (to x). As it is well known, every LUR space has property (H). A Banach space X is rotund (also called strictly convex) if every point x ∈ S X is extremal. Obviously, every MLUR space is rotund. Let C be a non-empty subset of a Banach space X and let x * ∈ S X * bounded above on C. We denote S(C, x * , δ) the δ-section defined by x * in C, i.e., S(C, x * , δ) := {x ∈ C; x, x * ≥ sup C x * − δ}.

In this paper we will give a sufficient condition for Dunford-Pettis property in Banach spaces. More precisely, if Banach space X has a basic, normalized system of vectors (xn), which is f (n)-approximate l1, then X has the Dunford-Pettis property.

Journal of Soviet Mathematics, 1986

Nonlinear Analysis: Theory, Methods & Applications, 2012

We study the stability properties of the class of weak*-extensible spaces introduced by Wang, Zhao, and Qiang showing, among other things, that weak*-extensibility is equivalent to having a weak*-sequentially continuous dual ball (in short, w*SC) for duals of separable spaces or twisted sums of w*SC spaces. This shows that weak*-extensibility is not a 3-space property, solving a question posed by Wang, Zhao, and Qiang. We also introduce a restricted form of weak*-extensibility, called separable weak*-extensibility, and show that separably weak*-extensible Banach spaces have the Gelfand-Phillips property, although they are not necessarily w*SC spaces.