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
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