More gateaux differentiability spaces

1985 ◽  
pp. 186-199
Author(s):  
C. Stegall
Keyword(s):  
Gâteaux Differentiability ◽  
Gateaux Differentiability

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.

Sign in / Sign up

Export Citation Format

Share Document