Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces

1999 ◽  
Vol 51 (1) ◽  
pp. 26-48 ◽  
Author(s):  
Marián Fabian ◽  
Boris S. Mordukhovich
Keyword(s):  
Banach Spaces ◽  
Reduction Method ◽  
Direct Proof ◽  
Density Theorem ◽  
Asplund Spaces ◽  
Separable Subspace ◽  
Separable Case

AbstractWe develop a method of separable reduction for Fréchet-like normals and ε-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fréchet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of ε-normals.

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 Properties of the trajectories of set-valued integrals in banach spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 271-281
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Mosco Convergence ◽  
Convexity Theorem ◽  
Convergence Theorems ◽  
Convergence Of Sets ◽  
Differentiability Properties

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

scholarly journals ON λ-STRICT IDEALS IN BANACH SPACES

2010 ◽  
Vol 83 (2) ◽  
pp. 231-240 ◽  
Author(s):  
TROND A. ABRAHAMSEN ◽  
OLAV NYGAARD
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Separable Case ◽  
Baire One ◽  
Ideal Property ◽  
Open Question

AbstractWe define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

Smooth Banach spaces, weak asplund spaces and monotone or usco mappings

1990 ◽  
Vol 72 (3) ◽  
pp. 257-279 ◽  
Author(s):  
David Preiss ◽  
R. R. Phelps ◽  
I. Namioka
Keyword(s):  
Banach Spaces ◽  
Asplund Spaces

scholarly journals Quasi reflexivity and the sup of linear functionals

1997 ◽  
Vol 55 (1) ◽  
pp. 89-98
Author(s):  
P.K. Jain ◽  
K.K. Arora ◽  
D.P. Sinha
Keyword(s):  
Banach Spaces ◽  
Unit Sphere ◽  
Linear Functionals ◽  
Reflexive Banach Spaces ◽  
Bounded Linear ◽  
Asplund Spaces ◽  
Linearly Independent

Quasi reflexive Banach spaces are characterised among the weakly countably determined Asplund spaces, in terms of the cardinality of the sets of linearly independent bounded linear functionals each of which does not attain its supremum on the unit sphere.

On Convex Functions Having Points of Gateaux Differentiability Which are Not Points of Fréchet Differentiability

1993 ◽  
Vol 45 (6) ◽  
pp. 1121-1134 ◽  
Author(s):  
J. M. Borwein ◽  
M. Fabian
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Infinite Dimensional ◽  
Gâteaux Differentiability ◽  
Hadamard Differentiability ◽  
Equivalent Norms ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
Gateaux Differentiability ◽  
Made In

AbstractWe study the relationships between Gateaux, Fréchet and weak Hadamard differentiability of convex functions and of equivalent norms. As a consequence we provide related characterizations of infinite dimensional Banach spaces and of Banach spaces containing ł1. Explicit examples are given. Some renormings of WCG Asplund spaces are made in this vein.

ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES

2001 ◽  
Vol 03 (04) ◽  
pp. 615-647 ◽  
Author(s):  
HEINZ H. BAUSCHKE ◽  
JONATHAN M. BORWEIN ◽  
PATRICK L. COMBETTES
Keyword(s):  
Banach Spaces ◽  
Reflexive Banach Space ◽  
Legendre Function ◽  
Fundamental Property ◽  
Strict Convexity ◽  
Directional Derivatives ◽  
Legendre Functions ◽  
Bregman Distances ◽  
Asplund Spaces ◽  
New Formula

The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed and several useful characterizations are given. The proofs rely on new results on the more subtle behavior of subdifferentials and directional derivatives at boundary points of the domain. In weak Asplund spaces, a new formula allows the recovery of the subdifferential from nearby gradients. Finally, it is shown that every Legendre function on a reflexive Banach space is zone consistent, a fundamental property in the analysis of optimization algorithms based on Bregman distances. Numerous illustrating examples are provided.

scholarly journals FLAT SETS, ℓp-GENERATING AND FIXING c0 IN THE NONSEPARABLE SETTING

2009 ◽  
Vol 87 (2) ◽  
pp. 197-210
Author(s):  
M. FABIAN ◽  
A. GONZÁLEZ ◽  
V. ZIZLER
Keyword(s):  
Banach Spaces ◽  
Alternative Proof ◽  
Asplund Spaces ◽  
Compactly Generated

AbstractWe define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or ℓp(ω1)-generated, where p∈(1,∞). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of ℓp(ω1) is ℓp(ω1)-generated for every p∈(1,∞). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal’s theorem on fixing c0(ω1).

scholarly journals Comparable differentiability characterisations of two classes of Banach spaces

1997 ◽  
Vol 56 (2) ◽  
pp. 263-272 ◽  
Author(s):  
J.R. Giles
Keyword(s):  
Banach Spaces ◽  
Equivalent Norm ◽  
Differentiability Property ◽  
Asplund Spaces

We characterise Banach spaces not containing l1 by a differentiability property of each equivalent norm and show that a slightly stronger differentiability property characterises Asplund spaces.

scholarly journals Comparable characterisations of two classes of Banach spaces by subdifferentials

1998 ◽  
Vol 58 (1) ◽  
pp. 155-158 ◽  
Author(s):  
J.R. Giles
Keyword(s):  
Banach Spaces ◽  
Equivalent Norms ◽  
Asplund Spaces ◽  
Continuity Properties

We characterise Banach spaces not containing ℓ1 and Banach spaces which are Asplund spaces by continuity properties of the subdifferential mappings of their equivalent norms.

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