Some properties which generalize uniform rotundity of Banach spaces (original) (raw)

On 𝑃- and 𝑝-Convexity of Banach Spaces

Abstract and Applied Analysis, 2010

We show that everyU-space and every Banach spaceXsatisfyingδX(1)>0areP(3)-convex, and we study the nonuniform version ofP-convexity, which we callp-convexity.

Some properties of C-reflexive locally convex spaces

Publikacije Elektrotehni?kog fakulteta - serija: matematika, 2007

In this note, we shall prove that the class of C-reflexive spaces is stable with respect to separated quotient, arbitrary product and sum, which is not the case for the closed subspaces and the dense hyperplanes. If the quotient mapping lifts compact disks, then the class of C-reflexive spaces is three-space stable.

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.

Kadec and Krein–Milman properties

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000

The main goal of this paper is to prove that any Banach space X with the Krein-Milman property such that the weak and the norm topology coincide on its unit sphere admits an equivalent norm that is locally uniformly rotund.

A note on the structure of WUR Banach spaces

We present an example of a Banach space E admitting an equivalent weakly uniformly rotund norm and such that there is no � : E ! c0(), for any set , linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hajek and Zizler. The space E is actually the dual space Yof a space Y which is a subspace of a WCG space.

Uniformly convex Banach spaces are reflexive—constructively

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive. Our aim in this note is to present a fully constructive 1 analysis of the Milman-Pettis theorem [11, 12, 9, 13]: a uniformly convex Banach space is reflexive. First, though, we need to outline the constructive context of the statement and proof of this theorem, and to clarify the terms we will use. The context of our work is that of a quasinormed space: that is, a linear space X equipped with a family (i) i∈I of seminorms on X such that the subset {{x i : i ∈ I} of R is bounded; that family is called the quasinorm on X. For all x, x in the quasinormed space X we define the inequality and equality relations by: • (x = x) ≡ ∃ i∈I (x − x i > 0); and • (x = x) ≡ ∀ i∈I (x − x i = 0) .