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A A -differentiability and A A -analyticity
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Journal Article
A A -differentiability and A A -analyticity
1996
Overview
Let AA be a finite-dimensional commutative algebra over R\\mathbb {R} and let CAr(U)C_{A}^{r}(U), Cω(U,A)C^{\\omega }(U,A) and OA(U)\\mathcal { O}_{A}(U) be the ring of AA-differentiable functions of class Cr,0≤r≤∞C^{r},\\,0 \\leq r \\leq \\infty, the ring of real analytic mappings with values in AA and the ring of AA-analytic functions, respectively, defined on an open subset UU of AnA^{n}. We prove two basic results concerning AA-differentiability and AA-analyticity: 1st1^{st}) OA(U)=CA∞(U)⋂Cω(U,A)\\mathcal { O}_{A}(U) = C^{\\infty }_{A}(U) \\bigcap C^{\\omega }(U,A), 2nd2^{nd}) OA(U)=CA∞(U)\\mathcal { O}_{A}(U) = C^{\\infty }_{A}(U) if and only if AA is defined over C\\mathbb {C}.
Publisher
American Mathematical Society
Subject
