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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)$
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Joram Lindenstrauss, David Preiss, and Jaroslav Tier

Print publication date: 2012

Print ISBN-13: 9780691153551

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153551.001.0001

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Smoothness and Asymptotic Smoothness

Smoothness and Asymptotic Smoothness

(p.133) Chapter Eight Smoothness and Asymptotic Smoothness
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Joram Lindenstrauss

David Preiss

Tiˇser Jaroslav

Princeton University Press

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 of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

Keywords:   modulus, subspace, Fréchet differentiability, asymptotic uniform smoothness, smooth bump, asymptotically smooth norm, Lipschitz function, renorming

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