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For each k ∈ N, we describe a mapping fk : C → Ek into a suitable non-complete complex locally convex space Ek such that fk is k times continuously complex differentiable (i.e., a Ck C -map) but not C C and hence not complex analytic. We also describe a complex analytic map from l1 to a suitable complete complex locally convex space E which is unbounded on each non-empty open subset of l1. Finally, we present a smooth map R → E into a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. Classification: 46G20 (primary), 26E05, 26E15, 26E20, 46T25

Instructive examples of smooth complex differentiable and complex analytic mappings into locally convex spaces