Nonlinear mappings in locally convex spaces

Abstract:

A notion of local linear approximation is defined for a nonlinear mapping, $f$, defined on one locally convex linear topological space with values in another. By use of this notion, theorems on the local solvability of the equation $y = f(x)$ and on the existence of a local inverse for $f$ are obtained. The continuity and linear approximability of the inverse are discussed. In addition, a relationship between the notion of linear approximation used in the paper and the notion of Fréchet differentiability is shown in the case the intervening spaces are Banach spaces.