## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Ultrametric Cn-Spaces of Countable Type

W.H. Schikhof

#### Abstract

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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 14, Number 5 (2007), 993-1000.

Dates
First available in Project Euclid: 17 December 2007

https://projecteuclid.org/euclid.bbms/1197908909

Digital Object Identifier
doi:10.36045/bbms/1197908909

Mathematical Reviews number (MathSciNet)
MR2379003

Zentralblatt MATH identifier
1134.46050

#### Citation

Schikhof, W.H. Ultrametric C n -Spaces of Countable Type. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 5, 993--1000. doi:10.36045/bbms/1197908909. https://projecteuclid.org/euclid.bbms/1197908909