Comparison of some notions of Ck-maps in multi-variable non-archimedian analysis

Abstract

Various definitions of $C^k$-maps on open subsets of finite-dimensional vector spaces over a complete valued field have been proposed in the literature. We show that the $C^k$-maps considered by Schikhof and De Smedt coincide with those of Bertram, Glöckner and Neeb. By contrast, Ludkovsky's $C^k$-maps need not be $C^k$ in the former sense, at least in positive characteristic. We also compare various types of Hölder differentiable maps on finite-dimensional and metrizable spaces.