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Transactions of the American Mathematical Society

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Infinitesimal calculus on locally convex spaces. I. Fundamentals


Author: K. D. Stroyan
Journal: Trans. Amer. Math. Soc. 240 (1978), 363-383
MSC: Primary 46G05; Secondary 02H25, 26A98, 58C20
DOI: https://doi.org/10.1090/S0002-9947-1978-0493323-4
MathSciNet review: 0493323
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Abstract: Differential calculus on nonnormed locally convex spaces suffers from technical difficulties (and the subsequent plethora of different definitions) partly because the families of multilinear maps over the spaces do not inherit a suitable topology. In this note we give the elementary ingredients of a strong differentiation based on Abraham Robinson's theory of infinitesimals.

Though nontopologizable, our theory is a natural generalization of standard infinitesimal calculus (finite dimensional or Banach space), see Robinson [1966], Keisler [1976], or Stroyan and Luxemburg [1976]. It is simpler than many recent developments, e.g., Yamamuro [1974] and Keller [1974]. The technical improvement of our approach should lead to advances in a variety of subjects.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0493323-4
Keywords: Infinitesimal calculus, locally convex space
Article copyright: © Copyright 1978 American Mathematical Society