Smoothness and Asymptotic Smoothness

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Modulus Of Smoothness ◽  
Lipschitz Functions ◽  
New Approach ◽  
Uniformly Smooth ◽  
Uniform Smoothness ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

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.

Ε‎-Fr ´Echet Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Simple Proof ◽  
Common Point ◽  
Lipschitz Functions ◽  
Closed Convex Hull ◽  
Uniformly Smooth ◽  
Fréchet Differentiability ◽  
Finite Set ◽  
Frechet Differentiability

This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

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.

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.

Fr ´Echet Differentiability of Vector-Valued Functions

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Mean Value ◽  
Common Point ◽  
Lipschitz Functions ◽  
Lipschitz Map ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Special Case

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

Introduction

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principles ◽  
Mean Value ◽  
Lipschitz Functions ◽  
Asplund Space ◽  
Lipschitz Mapping ◽  
Countable Collection ◽  
Fréchet Differentiability ◽  
Nikodym Property ◽  
Frechet Differentiability ◽  
And Variational Principles

This book deals with the existence of Fréchet derivatives of Lipschitz functions from X to Y, where X is an Asplund space and Y has the Radon-Nikodým property (RNP). It considers whether every countable collection of real-valued Lipschitz functions on an Asplund space has a common point of Fréchet differentiability. It also examines the conditions under which all Lipschitz mapping of X to finite dimensional spaces not only possess points of Fréchet differentiability, but possess so many of them that even the multidimensional mean value estimate holds. Other topics include the notion of the Radon-Nikodým property and main results on Gâteaux differentiability of Lipschitz functions and related notions of null sets; separable determination and variational principles; and differentiability of Lipschitz maps on Hilbert spaces.

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