Chapter

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 by Princeton University Press

Published in print February 2012 | ISBN: 9780691153551
Published online October 2017 | e-ISBN: 9781400842698
Smoothness, Convexity, Porosity, and Separable Determination

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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 discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

Keywords: separable dual; Fréchet smooth norm; convex function; Banach space; renorming; Fréchet differentiability; porous sets; separable determination; nonseparable space

Chapter.  8604 words. 

Subjects: Mathematics

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