Fréchet differentiability of Lipschitz maps and porous sets in Banach spaces

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Jaroslav Tišer
Keyword(s):  
Banach Spaces ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets

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.

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.

Porosity and ε‎-Fr échet differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Porous Set ◽  
Delta Set ◽  
Lipschitz Maps ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Null Set

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Jaroslav Tier
Keyword(s):  
Banach Spaces ◽  
Set Theory ◽  
Lipschitz Functions ◽  
Descriptive Set Theory ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Descriptive Set ◽  
Vector Valued ◽  
Derivatives Of

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Unavoidable Porous Sets and Nondifferentiable Maps

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Mean Value ◽  
Porous Set ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Derivatives Of

This chapter discusses Γ‎ₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ‎-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε‎-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γ‎ₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.

A Dual characterization of Banach Spaces With the Convex Point-of-Continuity Property

1989 ◽  
Vol 32 (3) ◽  
pp. 274-280
Author(s):  
D. E. G. Hare
Keyword(s):  
Banach Spaces ◽  
Continuity Property ◽  
Dual Spaces ◽  
Equivalent Norms ◽  
Fréchet Differentiability ◽  
New Type ◽  
Frechet Differentiability ◽  
Point Of Continuity Property ◽  
Convex Point

AbstractWe introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

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