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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

## 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

## 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

##
*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

## Topological Degree Computation

*Krzysztof A. Sikorski*

### in Title Pages

- Published in print:
- 2001
- Published Online:
- November 2020
- ISBN:
- 9780195106909
- eISBN:
- 9780197561010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195106909.003.0008
- Subject:
- Computer Science, Mathematical Theory of Computation

In this chapter we address the problem of computing topological degree of Lipschitz functions. From the knowledge of the topological degree one may ascertain whether ... More

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