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.

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 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.

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.

scholarly journals On Weighted Polynomial Approximation With Gaps

2005 ◽  
Vol 178 ◽  
pp. 55-61 ◽  
Author(s):  
Guantie Deng
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Polynomial Approximation ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Weighted Polynomial Approximation ◽  
Necessary And Sufficient Condition ◽  
Weighted Banach Space ◽  
Nonnegative Continuous Function ◽  
Necessary And Sufficient

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

Smooth Partitions of Unity on Banach Spaces

1998 ◽  
Vol 41 (2) ◽  
pp. 145-150
Author(s):  
R. Fry
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bump Function

AbstractIt is shown that if a Banach space X admits a Ck-smooth bump function, and X* is Asplund, then X admits Ck-smooth partitions of unity.

Sign in / Sign up

Export Citation Format

Share Document