Measures of Weak Noncompactness in Non-Archimedean Banach Spaces

Archiv der Mathematik, 2013

This concept, introduced independently by Granero (2006) and Cascales-Marciszewski-Raja (2006), has been used to study a quantitative version of Krein's theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper a quantitative version of Krein's theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E the above function k(H) reads as follows k(H) := sup{d (h, E) : h ∈ H σ(E ,E) }, where d(h, E) is the natural distance of h to E in the bidual E. The main result of the paper is the following Theorem: For a bounded set H in a Fréchet space E the following inequality holds k(coH) < (2 n+1 − 2)k(H) + 1 2 n for all n ∈ N. Consequently this yields also the following formula k(coH) ≤ k(H)(3 − 2 k(H)). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein's theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski and Raja) to the class of Fréchet space. We also define and discuss two another measures of weak non-compactness lk(H) and k (H) for a Fréchet space and provide two quantitative versions of Krein's theorem for the both functions.

Revista Matemática Iberoamericana, 2000

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

2020

Based on the notion of ν -convergence of bounded linear operators defined by A. Mario in [3], we introduce this convergence in a non-Archimedean Banach space and we study its properties. Besides, we introduce the new notion of collectively compact convergence in a non-Archimedean setting.

Journal of Convex Analysis, 2018

Let K be a non-archimedean valued field and let E be a non-archimedean Banach space over K. By Ew we denote the space E equipped with its weak topology and by E∗ w∗ the dual space E ∗ equipped with its weak∗ topology. Several results about countable tightness and the Lindelöf property for Ew and E ∗ w∗ are provided. A key point is to prove that for a large class of infinite-dimensional polar Banach spaces E, countable tightness of Ew or E ∗ w∗ implies separability of K. As a consequence we obtain the following two characterizations of the field K: (a) A non-archimedean valued field K is locally compact if and only if for every Banach space E over K the space Ew has countable tightness if and only if for every Banach space E over K the space E∗ w∗ has the Lindelöf property. (b) A non-archimedean valued separable field K is spherically complete if and only if every Banach space E over K for which Ew has the Lindelöf property must be separable if and only if every Banach space E over K...

Mathematische Annalen, 2004

Suppose ∞ → X. We construct examples of bounded sets M ⊂ X, such that M w * ⊂ X + 1 2 B X * , but coM w * ⊂ X + αB X * * for any α < 1. These examples show that the previous results of the authors on quantitative versions of Krein's theorem are optimal.

Journal of Mathematical Analysis and Applications, 1991

Journal of Convex Analysis

A point x ∈ A ⊂ (X, · ) is quasi-denting if for every ε > 0 there exists a slice of A containing x with Kuratowski index less than ε. The aim of this paper is to generalize the following theorem with a geometric approach, see : A Banach space such that every point of the unit sphere is quasi-denting (for the unit ball) admits an equivalent LUR norm.

Acta Mathematica Hungarica, 2010

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.

Linear Algebra and its Applications, 2012

Throughout this paper a study on the Krein-Milmam Property and the Bade Property is entailed reaching the following conclusions: If a real topological vector space satisfies the Krein-Milmam Property, then it is Hausdorff; if a real topological vector space satisfies the Krein-Milmam Property and is locally convex and metrizable, then all of its closed infinite dimensional vector subspaces have uncountable dimension; if a real pseudo-normed space has the Bade Property, then it is Hausdorff as well but could allow closed infinite dimensional vector subspaces with countable dimension. On other hand, we show the existence of infinite dimensional closed subspaces of ∞ with the Bade Property that are not the space of convergence associated to any series in a real topological vector space. Finally, we characterize unconditionally convergent series in real Banach spaces by means of a new concept called uniform convergence of series.

Archiv der Mathematik, 1994