Exponential laws for ultrametric partially differentiable functions and applications

2012/09/06 by Helge Glockner, Glockner, Helge
Mathematics · #12J25 #26B05 #26E20 #26E30 (Primary) 12J10 #30G06 #32P05 #54D50 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:12J10 #msc:12J25 #msc:26B05 #msc:26E20 #msc:26E30 #msc:30G06 #msc:32P05 #msc:54D50

paper · pdf · doi:10.48550/arxiv.1209.1384

73 pages, LaTeX, extended preprint version; v3: typos corrected

arxiv created 2012/09/11 · arxiv updated 2012/09/12

Abstract

We establish exponential laws for certain spaces of differentiable functions over a valued field K. For example, we show that the topological vector spaces C^r,s(U x V,E) and C^r(U,C^s(V,E)) are isomorphic if U and V are open subsets of Kⁿ and K^m, respectively, E is a topological K-vector space, and r,s are degrees of differentiability. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by E. Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C^r-functions on powers Z_pⁿ of the p-adic integers in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.

Related

• Exponential laws for spaces of differentiable functions on topological groups
• A p-adic Montel theorem and locally polynomial functions
• Smoothness of functions global and along curves over ultra-metric fields
• Exponential laws for weighted function spaces and regularity of weighted mapping groups
• Maps on Ultrametric Spaces, Hensel's Lemma, and Differential Equations Over Valued Fields