scholarly journals Asplund spaces and a variant of weak uniform rotundity

2000 ◽  
Vol 61 (3) ◽  
pp. 451-454 ◽  
Author(s):  
John Giles ◽  
Jon Vanderwerff
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Uniform Rotundity ◽  
Rotund Norm ◽  
Rotund Banach Space

We introduce a property formally weaker than weak uniform rotundity, which we call equatorial weak uniform rotundity. We show that an equatorially weakly uniformly rotund norm need not be weakly locally uniformly rotund. Nevertheless, we show that an equatorially weakly uniformly rotund Banach space is an Asplund space.

scholarly journals A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 450-455
Author(s):  
J. R. GILES
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Weakly Compact ◽  
Asplund Spaces ◽  
Subdifferential Mapping

AbstractA Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

scholarly journals On the characterisation of Asplund spaces

1982 ◽  
Vol 32 (1) ◽  
pp. 134-144 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Direct Proof ◽  
Asplund Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Separable Subspace ◽  
Nikodym Property ◽  
Continuous Convex Function ◽  
Fréchet Differentiable ◽  
Open Convex Subset

AbstractA Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

scholarly journals On the weak uniform rotundity of Banach spaces

2003 ◽  
Vol 2003 (30) ◽  
pp. 1943-1945
Author(s):  
Wen D. Chang ◽  
Ping Chang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Product Space ◽  
Quotient Space ◽  
Uniform Rotundity ◽  
Rotund Banach Space

We prove that ifXi,i=1,2,…,are Banach spaces that are weak* uniformly rotund, then theirlpproduct space(p>1)is weak* uniformly rotund, and for any weak or weak* uniformly rotund Banach space, its quotient space is also weak or weak* uniformly rotund, respectively.

scholarly journals K-Uniform Rotundity of Sequence Orlicz Spaces

1991 ◽  
Vol 34 (1) ◽  
pp. 128-135 ◽  
Author(s):  
Wang Tingfu ◽  
Chen Shutao
Keyword(s):  
Banach Space ◽  
Orlicz Spaces ◽  
Uniform Rotundity ◽  
Rotund Banach Space

AbstractThis paper presents a criterion of KUR for sequence Orlicź spaces with Luxemburg's norm. The result also indicates that for any integer k ≥ 1, there exists a k + 1-uniformly rotund Banach space not being k-uniformly rotund.

scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

scholarly journals The differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Inductive Limit ◽  
Asplund Space ◽  
Fréchet Spaces ◽  
Linear Spaces ◽  
Asplund Spaces ◽  
Continuous Convex Function ◽  
Mazur’S Theorem ◽  
Topological Linear ◽  
Differentiability Properties

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals Smoothness and the asymptotic-norming properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Asplund Space ◽  
Duality Mapping

We study some smoothness properties of a Banach space X that are related to the weak* asymptotic-norming properties of the dual space X*. These properties imply that X is an Asplund space and are related to the duality mapping of X.

scholarly journals A note on bump functions that locally depend on finitely many coordinates

1997 ◽  
Vol 56 (3) ◽  
pp. 447-451 ◽  
Author(s):  
M. Fabian ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Linear Span ◽  
Bump Function ◽  
Closed Linear Span ◽  
Asplund Spaces

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

Sign in / Sign up

Export Citation Format

Share Document