Fr ´Echet Differentiability Except For Γ-Null Sets
Fr ´Echet Differentiability Except For Γ-Null Sets
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 negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.
Keywords: infinite dimensional space, Fréchet differentiability, Γ-null sets, porous sets, Gâteaux derivative, Banach space, Lipschitz function, Lipschitz map
Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.
Please, subscribe or login to access full text content.
If you think you should have access to this title, please contact your librarian.
To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.