Differentials of Functions with Arguments and Values in Topological Abelian Groups
Michal, A. D. (1940) Differentials of Functions with Arguments and Values in Topological Abelian Groups. Proceedings of the National Academy of Sciences of the United States of America, 26 (5). pp. 356-359. ISSN 0027-8424. PMCID PMC1078188. https://resolver.caltech.edu/CaltechAUTHORS:20150811-112829955
By a topological abelian group T (t.a.g. T) we shall mean an abstract abelian group-written additively-such that (a) the function x + y and the inverse function -x are continuous functions (neighborhood continuity) of both variables x and y and of the variable x, respectively, with respect to a postulated Hausdorff topology; (b) given any y ε T and any Hausdorff neighborhood U of 0 ε T, there exists a "positive integer" n such that y ε nU. In this note we shall give brief indications of a differential calculus for functions f(x) with x e t.a.g. T_1 and values in a t.a.g. T_2. Proofs and further developments will appear elsewhere.
http://www.pnas.org/content/26/5/356.full.pdf Publisher Article
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078188/ PubMed Central Article
© 1940 National Academy of Sciences. Communicated April 15, 1940.
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Differentials of Functions with Arguments and Values in Topological Abelian Groups A. D. Michal PNAS 1940 26 (5) 356-359