**Graduate Studies in Mathematics**

Volume: 191;
2018;
466 pp;
Hardcover

MSC: Primary 46; 47;

Print ISBN: 978-1-4704-4190-6

Product Code: GSM/191

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-1-4704-4776-2
Product Code: GSM/191.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

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#### Supplemental Materials

# Functional Analysis

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*Theo Bühler; Dietmar A. Salamon*

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,
Kre&ibreve;n–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.

#### Readership

Graduate students and researchers interested in teaching and learning functional analysis.

#### Table of Contents

# Table of Contents

## Functional Analysis

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Introduction xi12
- Chapter 1. Foundations 116
- Chapter 2. Principles of Functional Analysis 4964
- Chapter 3. The Weak and Weak* Topologies 109124
- Chapter 4. Fredholm Theory 163178
- Chapter 5. Spectral Theory 197212
- Chapter 6. Unbounded Operators 295310
- Chapter 7. Semigroups of Operators 349364
- Appendix A. Zorn and Tychonoff 445460
- Bibliography 453468
- Notation 459474
- Index 461476
- Back Cover Back Cover1482