Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces
摘要
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.
引用
@article{arxiv.math/0701197,
title = {Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces},
author = {Helge Glockner},
journal= {arXiv preprint arXiv:math/0701197},
year = {2007}
}
备注
15 pages (v2: references for linear independence of exponential polynomials added; proof replaced by easier argument)