Introduction to Modern Analysis

Hardcover | November 7, 2003

byShmuel Kantorovitz

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This text is based on lectures given by the author at the advanced undergraduate and graduate levels in Measure Theory, Functional Analysis, Banach Algebras, Spectral Theory (of bounded and unbounded operators), Semigroups of Operators, Probability and Mathematical Statistics, and PartialDifferential Equations. The first 10 chapters discuss theoretical methods in Measure Theory and Functional Analysis, and contain over 120 end of chapter exercises. The final two chapters apply theory to applications in Probability Theory and Partial Differential Equations.The Measure Theory chapters discuss the Lebesgue-Radon-Nikodym theorem which is given the Von Neumann Hilbert space proof. Also included are the relatively advanced topics of Haar measure, differentiability of complex Borel measures in Euclidean space with respect to Lebesgue measure, and theMarcinkiewicz' interpolation theorem for operators between Lebesgue spaces. The Functional Analysis chapters cover the usual material on Banach spaces, weak topologies, separation, extremal points, the Stone-Weierstrass theorem, Hilbert spaces, Banach algebras, and Spectral Theory for both bounded and unbounded operators. Relatively advanced topics such as theGelfand-Naimark-Segal representation theorem and the Von Neumann double commutant theorem are included.The final two chapters deal with applications, where the measure theory and functional analysis methods of the first ten chapters are applied to Probability Theory and the Theory of Distributions and PDE's. Again, some advanced topics are included, such as the Lyapounov Central Limit theorem, theKolmogoroff "Three Series theorem", the Ehrenpreis-Malgrange-Hormander theorem on fundamental solutions, and Hormander's theory of convolution operators. The Oxford Graduate Texts in Mathematics series aim is to publish textbooks suitable for graduate students in mathematics and its applications. The level of books may range from some suitable for advanced undergraduate courses at one end, to others of interest to research workers. The emphasis is ontexts of high mathematical quality in active areas, particularly areas that are not well represented in the literature at present.

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From the Publisher

This text is based on lectures given by the author at the advanced undergraduate and graduate levels in Measure Theory, Functional Analysis, Banach Algebras, Spectral Theory (of bounded and unbounded operators), Semigroups of Operators, Probability and Mathematical Statistics, and PartialDifferential Equations. The first 10 chapters di...

Shmuel Kantorovitz is a Professor of Mathematics, Bar-Ilan Univerity, Israel.

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Format:HardcoverDimensions:446 pages, 9.21 × 6.14 × 1.11 inPublished:November 7, 2003Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198526563

ISBN - 13:9780198526568

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Table of Contents

Preface1. Measures2. Construction of Measures3. Measure and Topology4. Continuous Linear Functionals5. Duality6. Bounded Operators7. Banach Algebras8. Hilbert Spaces9. Intergral Representation10. Unbounded OperatorsApplication I:ProbabilityApplication II: DistributionsBibliographyIndex

Editorial Reviews

`This book is a valuable and pleasant contribution to the wide field of "modern analysis"'Harro Heuser, American Mathematical Society