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## 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

## 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

## 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

## 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

## 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

## 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

## LIPSCHITZ PATHS

*Terry Lyons and Zhongmin Qian*

### in System Control and Rough Paths

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198506485
- eISBN:
- 9780191709395
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506485.003.0002
- Subject:
- Mathematics, Probability / Statistics

The theory of controlled systems is worked out in detail for the case where the driving stimulus or noise is a Lipschitz function. The Itô functional, which is the map from stimulus to response, is ... 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

##
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

## 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

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