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An implicit function theorem in locally convex spaces. (English) Zbl 0562.46027

The notion of strict differentiability in Banach spaces is generalized to locally convex spaces in such a way, that computations in the bornological operator ideal of bounded operators play the role of norm estimations. We get a remarkably rich theory including not only the easy formulas like the chain rule, but also the deep theorems like the implicit function theorem. The Banach space proofs can be translated almost literally.

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
58C15 Implicit function theorems; global Newton methods on manifolds
47L10 Algebras of operators on Banach spaces and other topological linear spaces
46A08 Barrelled spaces, bornological spaces
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[1] [Russian Text Ignored]. 23 pp 67– (1968)
[2] Variétés différentielles et analytiques. Fascicule de résultats, [Russian Text Ignored]. 1–7, Hermann, Paris (1967)
[3] Differential calculus, Hermann/Kershaw, Paris/London (1971)
[4] Foundations of modern analysis, Academic Press, New York/London (1969)
[5] Hogbe-Nlend, Springer Lecture Notes in Math. 331 pp 84– (1973)
[6] Locally convex spaces and operator ideals, Teubner, Leipzig (1983) · Zbl 0552.46005
[7] Real analysis, second edition, Addison-Wesley, Reading/Mass. (1983)
[8] Yamamuro, Springer Lecture Notes in Math. 374 (1974) · Zbl 0276.58001 · doi:10.1007/BFb0061580
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