On the Gelfand-Phillips property in Banach spaces with PRI (original) (raw)

On a certain class of Banach spaces

Topology and its Applications, 2013

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.

On weakly countably determined Banach spaces

Transactions of the American Mathematical Society, 1987

ABSTRACT. For a topological space X, let Ci(X) denote the Banach space of all bounded functions /: X —► R such that for every e > 0 the set {x € X: \f{x)\ > e} is closed and discrete in X, endowed with the supremum norm. The main theorem is the following: Let L be a weakly countably ...

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.

Some strongly bounded classes of Banach spaces

Fundamenta Mathematicae, 2007

We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach spaces.

A New Class of Banach Spaces and Its Relation with Some Geometric Properties of Banach Spaces

Abstract and Applied Analysis, 2012

By introducing the concept ofL-limited sets and thenL-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.

On a certain class of Kσδ Banach spaces

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.

The strong Gelfand–Phillips property in Banach lattices

Banach Journal of Mathematical Analysis, 2016

We introduce the concept of the strong Gelfand-Phillips (GP) property in Banach lattices, and we characterize Banach lattices with the strong GP property. Next, by introducing the class of almost limited completely continuous operators from an arbitrary Banach lattice E to another F , we give some properties of them related to some well-known classes of operators and, especially, to the strong GP property of the Banach lattice E.