scholarly journals Generic Fréchet Differentiability of Convex Functions on Non-Asplund Spaces

1997 ◽  
Vol 214 (2) ◽  
pp. 367-377 ◽  
Author(s):  
Cheng Lixin ◽  
Shi Shuzhong ◽  
E.S. Lee
Keyword(s):  
Convex Functions ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

scholarly journals Separable determination of Fréchet differentiability of convex functions

1995 ◽  
Vol 52 (1) ◽  
pp. 161-167 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
Continuous Convex Function ◽  
Separability Condition ◽  
Frechet Differentiability ◽  
Open Convex Subset ◽  
Insight Into

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Fréchet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

Fr échet Differentiability of Real-Valued Functions

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principles ◽  
Mean Value ◽  
Lipschitz Functions ◽  
Monotone Functions ◽  
Porous Set ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
The Mean ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

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.

Monotone operators and dentability

1978 ◽  
Vol 18 (1) ◽  
pp. 77-82 ◽  
Author(s):  
S.P. Fitzpatrick
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
General Result ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Monotone Operators ◽  
Special Cases ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

P.S. Kenderov has shown that every monotone operator on an Asplund Banach space is continuous on a dense Gδ subset of the interior of its domain. We prove a general result which yields as special cases both Kenderov's Theorem and a theorem of Collier on the Fréchet differentiability of weak* lower semicontinuous convex functions.

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