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Differential calculus over general base fields and rings. (English) Zbl 1099.58006
The article is devoted to a generalization of classes of smoothness by W. H. Schikhof to functions of several variables and also including functions on topological vector spaces of infinite dimension over topological fields. This approach is applicable to the classical fields \(\mathbb R\) and \(\mathbb C\), as well as to fields with non-archimedean valuations. Additional variables related with linear shifts along vectors appearing in partial difference quotients of the \((n-1)\)-th order are included in the consideration while evaluation of \(C^n\) class of smoothness with the help of partial difference quotients of the \(n\)-th order by induction starting from the \(C^0\) concept. It permits to work effectively with classes of smoothness \(C^n\) and \(C^{\infty }\) over fields of non-zero characteristic with non-archimedean valuations. An analog of the Taylor formula is obtained, applications to Lie groups are considered, examples of \(C^n\) curves are given. In the case of infinite fields of zero characteristic with nontrivial non-archimedean valuation classes of smoothness \(C^n\) (with natural number \(n\)) on topological vector spaces considered in the paper under review are equivalent to classes of smoothness \(C^n\) of the papers of S. V. Ludkovsky [Ann. Math. Blaise Pascal 7, No. 2, 19–53 (2000; Zbl 0969.43002), Ann. Math. Blaise Pascal 7, No. 2, 55–80 (2000; Zbl 0972.43001), Theor. Math. Phys. 119, No. 3, 698–711 (1999; Zbl 0952.58008), Southeast Asian Bull. Math. 26, 975–1004 (2003; Zbl 1048.43006)]. It follows from the consideration of formulas for partial difference quotients of compositions of functions, since linear shift operators along vectors are \(C^{\infty }\) smooth on a given topological vector space.

MSC:
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
26E15 Calculus of functions on infinite-dimensional spaces
26E20 Calculus of functions taking values in infinite-dimensional spaces
26E30 Non-Archimedean analysis
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