Examples of differentiable mappings into non-locally convex spaces

Abstract

Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In particular, for a suitable non-locally convex space E, we describe a smooth injection of R into E whose derivative vanishes identically; we present a complex C^\infty-map on the complex field C which is not given locally by its Taylor series, around any point; we present a complex C^1-map into a complete, non-locally convex topological vector space which is not C^2; and we present a compactly supported, non-zero, complex C^\infty-map from C to a suitable non-locally convex space.