A Simple Theory of Differential Calculus in Locally Convex Spaces

Abstract

A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous $k$th order differentiability, smoothness of inverse functions, and function space continuity properties of higher order derivatives are examined. It is shown that this theory extends the classical Fréchet theory of differential calculus for maps between Banach spaces.