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" Functional analysis / "
Theo Bühler, Dietmar A. Salamon.
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BL
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| Record Number
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850300
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| Main Entry
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Bühler, Theo,1978-
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| Title & Author
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Functional analysis /\ Theo Bühler, Dietmar A. Salamon.
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| Publication Statement
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Providence, Rhode Island :: American Mathematical Society,, [2018]
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| Series Statement
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Graduate studies in mathematics ;; volume 191
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xiv, 466 pages ;; 27 cm.
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| ISBN
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147044190X
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: 9781470441906
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Includes bibliographical references and index.
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| Contents
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Introduction -- Foundations -- Principles of functional analysis -- The weak and weak* topologies -- Fredholm theory -- Spectral theory -- Unbounded operators -- Semigroups of operators.
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| Abstract
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Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics. This book provides a comprehensive introduction to the field for graduate students and researchers. It begins in Chapter 1 with an introduction to the necessary foundations, including the Arzela-Ascoli theorem, elementary Hilbert space theory, and the Baire Category Theorem. Chapter 2 develops the three fundamental principles of functional analysis (uniform boundedness, open mapping theorem, Hahn-Banach theorem) and discusses reflexive spaces and the James space. Chapter 3 introduces the weak and weak* topologies and includes the theorems of Banach-Alaoglu, Banach-Dieudonne, Eberlein-Smulyan, Krein-Milman, as well as an introduction to topological vector spaces and applications to ergodic theory. Chapter 4 is devoted to Fredholm theory. It includes an introduction to the dual operator and to compact operators, and it establishes the closed image theorem. Chapter 5 deals with the spectral theory of bounded linear operators. It introduces complex Banach and Hilbert spaces, the continuous functional calculus for self-adjoint and normal operators, the Gelfand spectrum, spectral measures, cyclic vectors, and the spectral theorem. Chapter 6 introduces unbounded operators and their duals. It establishes the closed image theorem in this setting and extends the functional calculus and spectral measure to unbounded self-adjoint operators on Hilbert spaces. Chapter 7 gives an introduction to strongly continuous semigroups and their infinitesimal generators. It includes foundational results about the dual semigroup and analytic semigroups, an exposition of measurable functions with values in a Banach space, and a discussion of solutions to the inhomogeneous equation and their regularity properties. The appendix establishes the equivalence of the Lemma of Zorn and the Axiom of Choice, and it contains a proof of Tychonoff's theorem. With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a one-or-two-semester course on functional analysis for beginning graduate students. Prerequisites are first-year analysis and linear algebra, as well as some foundational material from the second-year courses on point set topology, complex analysis in one variable, and measure and integration.
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Functional analysis.
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Semigroups of operators.
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Spectral theory (Mathematics)
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Functional analysis -- Instructional exposition (textbooks, tutorial papers, etc.).
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Functional analysis.
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Funktionalanalysis
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Operator theory -- Instructional exposition (textbooks, tutorial papers, etc.).
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Semigroups of operators.
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Spectral theory (Mathematics)
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| Dewey Classification
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515/.7
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| LC Classification
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QA320.B84 2018
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| NLM classification
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46-0147-01msc
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47-01.msc
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| Added Entry
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Salamon, D., (Dietmar)
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