scholarly journals Approximation of lipschitz functions by Δ-convex functions in banach spaces

1998 ◽  
Vol 106 (1) ◽  
pp. 269-284 ◽  
Author(s):  
Manuel Cepedello Boiso
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Lipschitz Functions

scholarly journals Continuity characterisations of differentiability of locally Lipschitz functions

1990 ◽  
Vol 41 (3) ◽  
pp. 371-380 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Lipschitz Functions ◽  
Distance Functions ◽  
Locally Lipschitz Functions ◽  
Locally Lipschitz ◽  
Differentiability Properties

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarke's non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.

Fr ´Echet Differentiability Except For Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Spaces ◽  
Common Point ◽  
Lipschitz Functions ◽  
Infinite Dimensional ◽  
Lipschitz Maps ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Gateaux Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

Banach Spaces of Lipschitz Functions

2019 ◽  
pp. 365-556
Author(s):  
Ştefan Cobzaş ◽  
Radu Miculescu ◽  
Adriana Nicolae
Keyword(s):  
Banach Spaces ◽  
Lipschitz Functions

scholarly journals On weak Hadamard differentiability of convex functions on Banach spaces

1996 ◽  
Vol 54 (1) ◽  
pp. 155-166 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Directional Differentiability ◽  
Hadamard Differentiability ◽  
Hadamard Directional Differentiability

We study two variants of weak Hadamard differentiability of continuous convex functions on a Banach space, uniform weak Hadamard differentiability and weak Hadamard directional differentiability, and determine their special properties on Banach spaces which do not contain a subspace topologically isomorphic to l1.

scholarly journals Approximation of Convex Functions on the Dual of Banach Spaces

2002 ◽  
Vol 116 (1) ◽  
pp. 126-140 ◽  
Author(s):  
Lixin Cheng ◽  
Yingbin Ruan ◽  
Yanmei Teng
Keyword(s):  
Banach Spaces ◽  
Convex Functions
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