On a certain class of Banach spaces (original) (raw)

2016

Using a strengthening of the concept of K σδ set, introduced in this paper, we study a certain subclass of the class of K σδ Banach spaces; the so called strongly K σδ Banach spaces. This class of spaces includes subspaces of strongly weakly compactly generated (SWCG) as well as Polish Banach spaces and it is related to strongly weakly Kanalytic (SWKA) Banach spaces as the known classes of K σδ and weakly K-analytic (WKA) Banach spaces are related.

A new class of weakly K-analytic Banach spaces

Commentationes Mathematicae Universitatis Carolinae

In this paper we define and investigate a new subclass of those Banach spaces which are K-analytic in their weak topology; we call them strongly weakly K-analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly K-analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show that: (i) not every separable Banach space is SWKA; (ii) every separable SWKA Banach space not containing ℓ 1 is Polish; (iii) we answer in the negative a question posed in [S-W] by constructing a subspace X of the SWCG space L 1 [0, 1] which is not SWCG.

On some new characterizations of weakly compact sets in Banach spaces

Studia Mathematica, 2010

In this paper 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 of X such that it admits the w2R-property on C; iv) there is a continuous and w * -lower semi-continuous (l.s.c.) semi-norm p on the dual X * with p ≥ sup C such that p 2 is everywhere Fréchet differentiable in X * ; and as a consequence, the space X is a weakly compactly generated (WCG) space if and only if there exists a continuous and w * -l.s.c. Fréchet smooth (not necessarily equivalent) norm on X * .

A CHARACTERIZATION OF SUBSPACES OF WEAKLY COMPACTLY GENERATED BANACH SPACES

Journal of the London Mathematical Society, 2004

We prove that a Banach space X is a subspace of a weakly com- pactly generated Banach space if and only if, for every&amp;amp;amp;amp;quot; &amp;amp;amp;amp;amp;gt; 0, X can be covered by a countable collection of bounded closed convex symmetric sets the weak closure in X of each of them lies within the distance&amp;amp;amp;amp;quot; from X. As a corollary, we give

Weak compactness and σ-Asplund generated Banach spaces

Studia Mathematica, 2007

σ-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results on Radon-Nikodým compacta. We show, typically, that in the framework of weakly Lindelöf determined Banach spaces, subspaces of weakly compactly generated spaces are the same as σ-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentiability, and of weak compactness in Banach spaces. As a consequence, we provide a functional-analytic proof of a result of Arvanitakis: A compact space is Eberlein if (and only if) it is simultaneously Corson and quasi-Radon-Nikodým.

Inner characterizations of weakly compactly generated Banach spaces and their relatives

Journal of Mathematical Analysis and Applications, 2004

We give characterizations of weakly compactly generated spaces, their subspaces, Vašák spaces, weakly Lindelöf determined spaces as well as several other classes of Banach spaces related to uniform Gâteaux smoothness, in terms of the presence of a total subset of the space with some additional properties. In addition, we describe geometrically, when possible, these classes by means of suitable smoothness or rotundity of the norm. As a consequence, we get new, functional analytic proofs of several theorems on (uniform) Eberlein, Gul'ko and Talagrand compacta.

Weakly K-analytic spaces and the three-space property for analyticity

Journal of Mathematical Analysis and Applications, 2010

First we show that the Mackey dual of a space C p (X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of C p (X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space N N. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.