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Porosity, Γ‎N- and 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/9780691153551.003.0010
Subject:
Mathematics, Analysis

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with ... 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


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


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 ´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


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


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