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

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

2007-01-06

arxiv.org/abs/math/0701197v2

Mathematics

Functional Analysis

15 pages (v2: references for linear independence of exponential polynomials added; proof replaced by easier argument)

Scientific paper

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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