Book:Mathematical Analysis II

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Bibliographical Data

Title:Mathematical Analysis II
Author:Elias Zakon
Key words:Lebesgue measure and integration, calculus in Banach spaces
Education Level:Higher Education
Description:The second volume of Zakon's Mathematical Analysis. The first chapter extends calculus to n-dimensional Euclidean space and, more generally, Banach spaces, covering the inverse function theorem, the implicit function theorem, Taylor expansions, etc. Some basic theorems in functional analysis, including the open mapping theorem and the Banach-Steinhaus uniform boundedness principle, are also proved. The text then moves to measure theory, discussing outer measures, Lebesgue measure, Lebesgue-Stieltjes measures, and differentiation of set functions. Fubini's theorem, the Radon-Nikodym theorem, and the basic convergence theorems (Fatou's lemma, the monotone convergence theorem, dominated convergence theorem) are covered. Finally, a chapter relates antidifferentiation to Lebesgue theory, Cauchy integrals, and convergence of parametrized integrals.


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