Spaces of Differentiable Functions and the Approximation Property

Abstract

Publisher Summary This chapter discusses spaces of differentiable functions and the approximation property. The chapter emphasizes a generalization of the theory of distributions to infinite dimensional locally convex (1.c) spaces by means of duality theory and with relations to infinite dimensional holomorphy. It studies spaces of differentiable functions on 1.c. spaces by analyzing the definitions of C n -functions. In the chapter, most of the definitions are stated along with some results. The chapter presents a sufficient condition for the approximation property. It also proves that for open subsets Ω 1 and Ω 2 in certain 1.c. spaces El and E2 respectively, there exists a natural topological isomorphism.