Differentiability and Related Properties of Nonlinear Operators: Some Aspects of the Role of Differentials in Nonlinear Functional Analysis

Abstract

This chapter discusses a few aspects of the role of differentials in nonlinear functional analysis. It presents a systematic exposition of several aspects of differential calculus in normed or topological linear spaces and highlights various settings in nonlinear functional analysis in which differentials play an important role. Differential splays at least three significant roles in the theory and application of nonlinear functional analysis. The first and the most familiar role is that of local approximation of a nonlinear map by a linear one. The differential is also important as a tool in nonlinear functional analysis. The third role of the differential is in the clue it provides for generalizations of a certain class of differentiable operators. The theory of differentiation in linear topological spaces is fairly complete for practical purpose.