Superanalysis, I. Differential calculus. (English. Russian original) Zbl 0552.46023

Theor. Math. Phys. 59, 317-335 (1984); translation from Teor. Mat. Fiz. 59, No. 1, 3-27 (1984).
The author studies differential calculus in superspaces over commutative Banach superalgebras. A Banach superalgebra is an associative Banach algebra \(\Lambda\) over the reals, with an identity, which is decomposed in a direct sum as a linear space, \(\Lambda =\Lambda_ 0\oplus \Lambda_ 1,\) admits a function \(p: \Lambda \to \{0,1\}\) such that \(p(a)=0\) for \(a\in \Lambda_ 0\), \(p(a)=1\) for \(a\in \Lambda_ 1\) and \(p(ab)=p(a)+p(b)(mod. 2).\) If \(ab-(-1)^{p(a)p(b)}=0\) then \(\Lambda\) is called a commutative superalgebra. The notion includes commutative Banach algebras and Grassmann algebras. A superspace of dimension (m,n) over \(\Lambda\) is a Banach space \[ R_{\Lambda}^{m,n}=\Lambda_ 0\times...\times \Lambda_ 0(m-times)\times \Lambda_ 1\times...\times \Lambda_ 1(n-times) \] with the norm \(\| (x_ 1,...,x_{m+n}\| =\sum^{m+n}_{i=1}\| x_ i\|\) and plays a role in superanalysis, analogous to that of \({\mathbb{R}}^ m\) in classical analysis. A superderivative is defined and a generalization of Cauchy-Riemann equations of complex analysis are derived. The basic rules of calculus are proved, including the chain rule, derivative of product, implicit function theorem and Taylor’s expansion. The rules hold also over the field of complex numbers, and some of them remain unchanged over arbitrary non-discrete complete field. The paper contains many examples and it is to be the first one in a sequence of the author’s papers on superanalysis.
Reviewer: A.Sterna-Karwat


46G05 Derivatives of functions in infinite-dimensional spaces
46H05 General theory of topological algebras
32H99 Holomorphic mappings and correspondences
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