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

Helge Glockner

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arXiv · math/0701197

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

Helge Glockner

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Abstract

For each positive integer k, we describe a map f from the complex plane to a suitable non-complete complex locally convex space such that f is k times continuously complex differentiable but not k+1 times, and hence not complex analytic. We also describe a complex analytic map from l^1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l^1. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduct, we find that free locally convex spaces over subsets of the complex plane with non-empty interior are not Mackey complete.

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Helge Glockner. 2007-04-28. Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces. https://arxiv.org/abs/math/0701197

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