Preliminaries to Main Results

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Flat Surface ◽  
Tensor Products ◽  
Linear Operators ◽  
Euclidean Spaces ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable ◽  
Derivatives Of ◽  
Surface Passing

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

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.

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.

scholarly journals Statistical expansions and locally uniform Fréchet differentiability

1991 ◽  
Vol 50 (1) ◽  
pp. 88-97 ◽  
Author(s):  
T. Bednarski ◽  
B. R. Clarke ◽  
W. Kolkiewicz
Keyword(s):  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable ◽  
Uniform Expansions ◽  
Uniform Expansion

AbstractEstimators which have locally uniform expansions are shown in this paper to be asymptotically equivalent to M-estimators. The M-functionals corresponding to these M-estimators are seen to be locally uniformly Fréchet differentiable. Other conditions for M-functionals to be locally uniformly Fréchet differentiable are given. An example of a commonly used estimator which is robust against outliers is given to illustrate that the locally uniform expansion need not be valid.

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.

A Non-Reflexive Smooth Space with a Smooth Dual

1974 ◽  
Vol 17 (4) ◽  
pp. 579-580
Author(s):  
J. H. M. Whitfield
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Gâteaux Differentiability ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Gateaux Differentiability ◽  
Fréchet Differentiable ◽  
Smooth Space ◽  
Nonreflexive Space

Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.

Smooth Norms in Orlicz Spaces

1991 ◽  
Vol 34 (1) ◽  
pp. 74-82 ◽  
Author(s):  
R. P. Maleev ◽  
S. L. Troyanski
Keyword(s):  
Orlicz Spaces ◽  
Equivalent Norm ◽  
Equivalent Norms ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable

AbstractEquivalent norms with best order of Frechet and uniformly Frechet differentiability in Orlicz spaces are constructed. Classes of Orlicz which admit infinitely many times Frechet differentiable equivalent norm are found.

Differentiability of Lipschitz Maps on Hilbert Spaces

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Fréchet Differentiability ◽  
Dimensional Version ◽  
Frechet Differentiability ◽  
Irregular Behavior

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with the theorem stating that every Lipschitz map of a Hilbert space to a two-dimensional space has points of Fréchet differentiability. This is followed by a lemma, which is stated in an arbitrary Hilbert space but whose validity in the general case follows from its three-dimensional version. The chapter then explains the proof of the theorem and of the lemma stated above. In particular, it considers two cases, one corresponding to irregular behavior and the other to regular behavior.

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