Abstract
For each k ∈ N, we describe a mapping $f_{k}:\mathbb{C} \longrightarrow E_{k}$ into a suitable non-complete complex locally convex space $E_{k}$ such that $f_{k}$ is $k$ times continuously complex differentiable (i.e., a $C^{k}_{\mathbb{C}}$-map) but not $C^{k+1}_{\mathbb{C}}$ and hence not complex analytic. We also describe a complex analytic map from $\ell^{1}$ to a suitable complete complex locally convex space $E$ which is unbounded on each non-empty open subset of $\ell^{1}$. Finally, we present a smooth map $\mathbb{R} \longrightarrow E$ into a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point.
Citation
Helge Glöckner. "Instructive examples of smooth complex differentiable and complex analytic mappings into locally convex spaces." J. Math. Kyoto Univ. 47 (3) 631 - 642, 2007. https://doi.org/10.1215/kjm/1250281028
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