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Hardcover ISBN:  9781470441906 
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Book DetailsGraduate Studies in MathematicsVolume: 191; 2018; 466 ppMSC: Primary 46; 47
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 Arzelà–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–Dieudonné, Eberlein–Šmulyan, Kreĭn–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 selfadjoint 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 selfadjoint 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 oneortwosemester course on functional analysis for beginning graduate students. Prerequisites are firstyear analysis and linear algebra, as well as some foundational material from the secondyear courses on point set topology, complex analysis in one variable, and measure and integration.
ReadershipGraduate students and researchers interested in teaching and learning functional analysis.

Table of Contents

Chapters

Foundations

Principles of functional analysis

The weak and weak* topologies

Fredholm theory

Spectral theory

Unbounded operators

Semigroups of operators

Zorn and Tychonoff


Additional Material

Reviews

The authors have done their best to write a book as accessible to students as possible...I think that [they] have achieved their goal and I can recommend this book.
Richard Becker, Mathematical Reviews 
This is a demanding book, but a valuable one.
Mark Hunacek, MAA Reviews


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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 Arzelà–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–Dieudonné, Eberlein–Šmulyan, Kreĭn–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 selfadjoint 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 selfadjoint 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 oneortwosemester course on functional analysis for beginning graduate students. Prerequisites are firstyear analysis and linear algebra, as well as some foundational material from the secondyear courses on point set topology, complex analysis in one variable, and measure and integration.
Graduate students and researchers interested in teaching and learning functional analysis.

Chapters

Foundations

Principles of functional analysis

The weak and weak* topologies

Fredholm theory

Spectral theory

Unbounded operators

Semigroups of operators

Zorn and Tychonoff

The authors have done their best to write a book as accessible to students as possible...I think that [they] have achieved their goal and I can recommend this book.
Richard Becker, Mathematical Reviews 
This is a demanding book, but a valuable one.
Mark Hunacek, MAA Reviews