2007/01/06 by Helge Glöckner, Glockner, Helge
Mathematics · #26E15 #26E20 #46G20 (primary) 26E05 #46T25 (secondary) #Advanced Banach Space Theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.math/0701197
openalex publication_date 2007/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 l1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l1. 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.