2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67962Examples 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.6 pagesFunctional Analysis58C20, 26E20, 46A16, 46G20text