Bulletin of the Belgian Mathematical Society - Simon Stevin

Ultrametric Cn-Spaces of Countable Type

W.H. Schikhof

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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

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

First available in Project Euclid: 17 December 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46S10: Functional analysis over fields other than $R$ or $C$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05]
Secondary: 26E30: Non-Archimedean analysis [See also 12J25]

non-archimedean Banach spaces differentiable functions spaces of countable type


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

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