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

*Joram Lindenstrauss, David Preiss, and Jaroslav Tier*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.001.0001
- Subject:
- Mathematics, Analysis

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 ... More

## Introduction

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0001
- Subject:
- Mathematics, Analysis

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 ... More

## Gâteaux differentiability of Lipschitz functions

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0002
- Subject:
- Mathematics, Analysis

This chapter presents the main results on Gâteaux differentiability of Lipschitz functions by recalling the notions of the Radon-Nikodým property (RNP) and null sets. The discussion focuses not only ... More

##
*Ε*-Fr ´Echet Differentiability

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0004
- Subject:
- Mathematics, Analysis

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 ... More

##
Γ-Null and Γ*N*-Null Sets

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0005
- Subject:
- Mathematics, Analysis

This chapter introduces the notions of Γ-null and Γₙ-null sets, which are σ-ideals of subsets of a Banach space X. Γ-null set is key for the strongest known general Fréchet differentiability ... More

## Preliminaries to Main Results

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0009
- Subject:
- Mathematics, Analysis

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 ... More

## Asymptotic Fr echet ´Differentiability

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0015
- Subject:
- Mathematics, Analysis

This chapter presents the current development of the first, unpublished proof of existence of points Fréchet differentiability of Lipschitz mappings to two-dimensional spaces. For functions into ... More

## Smoothness, Convexity, Porosity, and Separable Determination

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0003
- Subject:
- Mathematics, Analysis

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 ... More

## Fr ´Echet Differentiability Except For Γ-Null Sets

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0006
- Subject:
- Mathematics, Analysis

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 ... More

## Smoothness and Asymptotic Smoothness

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0008
- Subject:
- Mathematics, Analysis

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 ... More

##
Porosity and *ε*-Fr échet differentiability

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0011
- Subject:
- Mathematics, Analysis

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 ... More

## Fr échet Differentiability of Real-Valued Functions

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0012
- Subject:
- Mathematics, Analysis

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 ... More

## Fr ´Echet Differentiability of Vector-Valued Functions

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0013
- Subject:
- Mathematics, Analysis

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 ... More

## Unavoidable Porous Sets and Nondifferentiable Maps

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0014
- Subject:
- Mathematics, Analysis

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 ... More

## Differentiability of Lipschitz Maps on Hilbert Spaces

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0016
- Subject:
- Mathematics, Analysis

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 ... More

## Action-Minimizing Invariant Measures for Tonelli Lagrangians

*Alfonso Sorrentino*

### in Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0003
- Subject:
- Mathematics, Applied Mathematics

This chapter discusses the notion of action-minimizing measures, recalling the needed measure–theoretical material. In particular, this allows the definition of a first family of invariant sets, the ... More

## The Second-Order Master Equation

*Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions*

### in The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691190716
- eISBN:
- 9780691193717
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691190716.003.0005
- Subject:
- Mathematics, Applied Mathematics

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the ... More

## The Equilibrium Manifold: Postmodern Developments in the Theory of General Economic Equilibrium

*Yves Balasko*

- Published in print:
- 2009
- Published Online:
- August 2013
- ISBN:
- 9780262026543
- eISBN:
- 9780262255370
- Item type:
- book

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262026543.001.0001
- Subject:
- Economics and Finance, Econometrics

This book argues that, contrary to what many textbooks want readers to believe, the study of the general equilibrium model did not end with the existence and welfare theorems of the 1950s. These ... More

## Scattering Problems: A Variety of Topics

*G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos*

### in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691142173
- eISBN:
- 9781400842650
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142173.003.0006
- Subject:
- Mathematics, Applied Mathematics

This chapter continues the study of scattering problems in the case where the considered fields have harmonic time dependence and the involved chiral media are homogeneous. It begins by containing ... More

## Smoothness and uniform smoothness

*Kazimierz Goebel and Stanisław Prus*

### in Elements of Geometry of Balls in Banach Spaces

- Published in print:
- 2018
- Published Online:
- April 2019
- ISBN:
- 9780198827351
- eISBN:
- 9780191866265
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827351.003.0004
- Subject:
- Mathematics, Pure Mathematics

The notions of smoothness and uniform smoothness of a space are discussed. The relation with differentiability of the norm is shown. The main tool, the modulus of smoothness of a space is studied.

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