scholarly journals On Gâteaux Differentiability of Pointwise Lipschitz Mappings

2008 ◽  
Vol 51 (2) ◽  
pp. 205-216 ◽  
Author(s):  
Jakub Duda
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Monotone Function ◽  
Gâteaux Differentiability ◽  
Lipschitz Mappings ◽  
Gateaux Differentiability ◽  
Fréchet Differentiable ◽  
Set Of Points

AbstractWe prove that for every function f : X → Y , where X is a separable Banach space and Y is a Banach space with RNP, there exists a set A ∈ such that f is Gâteaux differentiable at all x ∈ S(f )\A, where S(f) is the set of points where f is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every K-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to ; this improves a result due to Borwein and Wang. Another corollary is that if X is Asplund, f : X → ℝ cone monotone, g : X → ℝ continuous convex, then there exists a point in X, where f is Hadamard differentiable and g is Fréchet differentiable.

A Non-Reflexive Smooth Space with a Smooth Dual

1974 ◽  
Vol 17 (4) ◽  
pp. 579-580
Author(s):  
J. H. M. Whitfield
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Gâteaux Differentiability ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Gateaux Differentiability ◽  
Fréchet Differentiable ◽  
Smooth Space ◽  
Nonreflexive Space

Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.

Γ‎-Null and Γ‎N-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Finite Dimension ◽  
Baire Category ◽  
Borel Sets ◽  
Basic Properties ◽  
Gâteaux Differentiability ◽  
Gateaux Differentiability ◽  
Null Set

This chapter introduces the notions of Γ‎-null and Γ‎ₙ-null sets, which are σ‎-ideals of subsets of a Banach space X. Γ‎-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γ‎ₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ‎-null and Γ‎ₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ‎-null and Γ‎ₙ-null sets of low Borel classes and presents equivalent definitions of Γ‎ₙ-null sets. Finally, it considers the separable determination of Γ‎-nullness for Borel sets.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

scholarly journals On Fixed Point Property under Lipschitz and Uniform Embeddings

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Jichao Zhang ◽  
Lingxin Bao ◽  
Lili Su
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Fixed Point Property ◽  
Point Property ◽  
Lipschitz Mappings ◽  
Affine Mappings ◽  
Reflexive Space ◽  
Gateaux Differentiability

We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.

scholarly journals Generic Gateaux differentiability via smooth perturbations

1997 ◽  
Vol 56 (3) ◽  
pp. 421-428
Author(s):  
Pando Gr Georgiev ◽  
Nadia P. Zlateva
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Bump Function ◽  
Space Applications ◽  
Gâteaux Differentiability ◽  
Gateaux Differentiability

We prove that in a Banach space with a Lipschitz uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gδ subset of the space, is Gateaux differentiable on a dense Gδ subset of the space. Applications of this result are given.

scholarly journals Structural implications of norms with Hölder right-hand derivatives.

1998 ◽  
Vol 57 (3) ◽  
pp. 415-425
Author(s):  
Michael O. Bartlett ◽  
John R. Giles ◽  
Jon D. Vanderwerff
Keyword(s):  
Banach Space ◽  
Equivalent Norm ◽  
Asplund Space ◽  
Hölder Condition ◽  
Gâteaux Differentiability ◽  
Right Hand ◽  
Gateaux Differentiability

We study a nonsmooth extension of Gateaux differentiability satisfying a directional Hölder condition. In particular, we show that a Banach space is an Asplund space if it has an equivalent norm with a directionally Hölder right-hand derivative at each point of its sphere.

New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals

1998 ◽  
Vol 41 (3) ◽  
pp. 279-289 ◽  
Author(s):  
Mariá D. Acosta ◽  
Manuel Ruiz Galán
Keyword(s):  
Banach Space ◽  
Positive Result ◽  
Unit Ball ◽  
Separable Banach Space ◽  
Intersection Property ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Norm Topology ◽  
Fréchet Differentiable ◽  
Norm Attaining

AbstractAs a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space X is very smooth or its bidual satisfies the w*-Mazur intersection property, then either X is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if X has the Mazur intersection property and so, if the norm of X is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

A Separable Banach Space with the Radon-Nikodym Property that is not Isomorphic to a Subspace of a Separable Dual

1980 ◽  
Vol 78 (1) ◽  
pp. 40 ◽  
Author(s):  
Philip W. McCartney ◽  
Richard C. O'Brien
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Nikodym Property
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