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December 2007 Ultrametric Cn-Spaces of Countable Type
W.H. Schikhof
Bull. Belg. Math. Soc. Simon Stevin 14(5): 993-1000 (December 2007). DOI: 10.36045/bbms/1197908909


Let $K$ be a non-trivially non-archimedean valued field that is complete with respect to the valuation $\left| \, \, \right| : K \longrightarrow[0,\infty)$, let $X$ be a non-empty subset of $K$ without isolated points. For $n \in \{ 0,1, \ldots \}$ the $K$-Banach space $BC^n(X)$, consisting of all $C^n$-functions $X \longrightarrow K$ whose difference quotients up to order $n$ are bounded, is defined in a natural way. It is proved that $BC^n(X)$ is of countable type if and only if $X$ is compact. In addition we will show that $BC^{\infty}(X) : = \bigcap _n BC^n(X)$, which is a Fréchet space with its usual projective topology, is of countable type if and only if $X$ is precompact.


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W.H. Schikhof. "Ultrametric Cn-Spaces of Countable Type." Bull. Belg. Math. Soc. Simon Stevin 14 (5) 993 - 1000, December 2007.


Published: December 2007
First available in Project Euclid: 17 December 2007

zbMATH: 1134.46050
MathSciNet: MR2379003
Digital Object Identifier: 10.36045/bbms/1197908909

Primary: ‎46S10
Secondary: 26E30

Keywords: differentiable functions , non-archimedean Banach spaces , spaces of countable type

Rights: Copyright © 2007 The Belgian Mathematical Society

Vol.14 • No. 5 • December 2007
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