Approximation and Gâteaux differentiability of convex function in Banach spaces

2021 ◽  
Author(s):  
Shaoqiang Shang ◽  
Yunan Cui
Keyword(s):  
Convex Function ◽  
Banach Spaces ◽  
Gâteaux Differentiability ◽  
Gateaux Differentiability

scholarly journals Convex spaces: Classification by differentiable convex functions

1992 ◽  
Vol 46 (1) ◽  
pp. 127-138 ◽  
Author(s):  
Roger Eyland ◽  
Bernice Sharp
Keyword(s):  
Convex Function ◽  
Banach Spaces ◽  
Convex Functions ◽  
Dual Space ◽  
Gâteaux Differentiability ◽  
Gateaux Differentiability ◽  
Open Question ◽  
Convex Spaces

The differentiability, of a specified strength, of a convex function at a point, is shown to be characterised by the convergence of subdifferentials in the appropriate topology on the dual space. This is used to prove that if each gauge is densely differentiable then so is each convex function. The generic version of this is equivalent to a conjecture which, for Gateaux differentiability and Banach spaces, is the long standing open question of whether X × ℝ is Weak Asplund whenever X is. Some progress is made towards a resolution.

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.

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.

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