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A First Course in Functional Analysis

ISBN: 978-0-470-14619-4
308 pages
April 2008
A First Course in Functional Analysis (0470146192) cover image
A concise introduction to the major concepts of functional analysis

Requiring only a preliminary knowledge of elementary linear algebra and real analysis, A First Course in Functional Analysis provides an introduction to the basic principles and practical applications of functional analysis. Key concepts are illustrated in a straightforward manner, which facilitates a complete and fundamental understanding of the topic.

This book is based on the author's own class-tested material and uses clear language to explain the major concepts of functional analysis, including Banach spaces, Hilbert spaces, topological vector spaces, as well as bounded linear functionals and operators. As opposed to simply presenting the proofs, the author outlines the logic behind the steps, demonstrates the development of arguments, and discusses how the concepts are connected to one another. Each chapter concludes with exercises ranging in difficulty, giving readers the opportunity to reinforce their comprehension of the discussed methods. An appendix provides a thorough introduction to measure and integration theory, and additional appendices address the background material on topics such as Zorn's lemma, the Stone-Weierstrass theorem, Tychonoff's theorem on product spaces, and the upper and lower limit points of sequences. References to various applications of functional analysis are also included throughout the book.

A First Course in Functional Analysis is an ideal text for upper-undergraduate and graduate-level courses in pure and applied mathematics, statistics, and engineering. It also serves as a valuable reference for practitioners across various disciplines, including the physical sciences, economics, and finance, who would like to expand their knowledge of functional analysis.

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Preface.

1. Linear Spaces and Operators.

1.1 Introduction.

1.2 Linear Spaces.

1.3 Linear Operators.

1.4 Passage from Finite- to Infinite-Dimensional Spaces.

Exercises.

2. Normed Linear Spaces: The Basics.

2.1 Metric Spaces.

2.2 Norms.

2.3 Space of Bounded Functions.

2.4 Bounded Linear Operators.

2.5 Completeness.

2.6 Comparison of Norms.

2.7 Quotient Spaces.

2.8 Finite-Dimensional Normed Linear Spaces.

2.9 Lp Spaces.

2.10 Direct Products and Sums.

2.11 Schauder Bases.

2.12 Fixed Points and Contraction Mappings.

Exercises.

3. Major Banach Space Theorems.

3.1 Introduction.

3.2 Baire Category Theorem.

3.3 Open Mappings.

3.4 Bounded Inverses.

3.5 Closed Linear Operators.

3.6 Uniform Boundedness Principle.

Exercises.

4. Hilbert Spaces.

4.1 Introduction.

4.2 Semi-Inner Products.

4.3 Nearest Points and Convexity.

4.4 Orthogonality.

4.5 Linear Functionals on Hilbert Spaces.

4.6 Linear Operators on Hilbert Spaces.

4.7 Order Relation on Self-Adjoint Operators.

Exercises.

5. Hahn–Banach Theorem.

5.1 Introduction.

5.2 Basic Version of Hahn–Banach Theorem.

5.3 Complex Version of Hahn–Banach Theorem.

5.4 Application to Normed Linear Spaces.

5.5 Geometric Versions of Hahn–Banach Theorem.

Exercises.

6. Duality.

6.1 Examples of Dual Spaces.

6.2 Adjoints.

6.3 Double Duals and Reflexivity.

6.4 Weak and Weak* Convergence.

Exercises.

7. Topological Linear Spaces.

7.1 Review of General Topology.

7.2 Topologies on Linear Spaces.

7.3 Linear Functionals on Topological Linear Spaces.

7.4 Weak Topology.

7.5 Weak* Topology.

7.6 Extreme Points and Krein–Milman Theorem.

7.7 Operator Topologies.

Exercises.

8. The Spectrum.

8.1 Introduction.

8.2 Banach Algebras.

8.3 General Properties of the Spectrum.

8.4 Numerical Range.

8.5 Spectrum of a Normal Operator.

8.6 Functions of Operators.

8.7 Brief Introduction to C-Algebras.

Exercises.

9. Compact Operators.

9.1 Introduction and Basic Definitions.

9.2 Compactness Criteria in Metric Spaces.

9.3 New Compact Operators from Old.

9.4 Spectrum of a Compact Operator.

9.5 Compact Self-Adjoint Operators on Hilbert Spaces.

9.6 Invariant Subspaces.

Exercises.

10. Application to Integral and Differential Equations.

10.1 Introduction.

10.2 Integral Operators.

10.3 Integral Equations.

10.4 Second-Order Linear Differential Equations.

10.5 Sturm–Liouville Problems.

10.6 First-Order Differential Equations.

Exercises.

11. Spectral Theorem for Bounded, Self-Adjoint Operators.

11.1 Introduction and Motivation.

11.2 Spectral Decomposition.

11.3 Extension of Functional Calculus.

11.4 Multiplication Operators.

Exercises.

Appendix A Zorn's Lemma.

Appendix B Stone–Weierstrass Theorem.

B.1 Basic Theorem.

B.2 Nonunital Algebras.

B.3 Complex Algebras.

Appendix C Extended Real Numbers and Limit Points of Sequences.

C.1 Extended Reals.

C.2 Limit Points of Sequences.

Appendix D Measure and Integration.

D.1 Introduction and Notation.

D.2 Basic Properties of Measures 258

D.3 Properties of Measurable Functions.

D.4 Integral of a Nonnegative Function.

D.5 Integral of an Extended Real-Valued Function.

D.6 Integral of a Complex-Valued Function.

D.7 Construction of Lebesgue Measure on R.

D.8 Completeness of Measures.

D.9 Signed and Complex Measures.

D.10 Radon–Nikodym Derivatives.

D.11 Product Measures.

D.12 Riesz Representation Theorem.

Appendix E Tychonoff's Theorem.

Symbols.

References.

Index.

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S. David Promislow, PhD, is Professor Emeritus of Mathematics at York University in Toronto, Canada. Dr. Promislow has over thirty-five years of teaching experience in the areas of functional analysis, group theory, measure theory, and actuarial mathematics. He is the author of Fundamentals of Actuarial Mathematics, also published by Wiley.
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  • Presents a unique coverage of the essential topics of functional analysis, based on successful lecture notes used by the author, for a concise one-semester course.
  • Explains the theories behind the concepts, taking the reader through the logic behind each argument’s development.
  • Features an array of exercises throughout the book, ranging from introductory to advanced, to enhance the reader’s learning experience.
  • Requires minimal previous knowledge and no background in either measure theory or general topology.
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"Graduate and advanced undergraduate students in mathematics and physics will appreciate this book as a useful and stimulating contribution to the vast array of textbooks on the subject.." (Zentralblatt MATH, October 2010)

"A First Course in Functional Analysis is an ideal text for upper-undergraduate and graduate-level courses in pure and applied mathematics, statistics, and engineering. It also serves as a valuable reference for practioners across various disciplines, including the physical sciences, economics, and finance, who would like to expand their knowledge of functional analysis." (Mathematical Reviews, 2009c)

"It is written in a very open, nontelegraphic style, and takes care to explain topics as they come up.  Recommended." (CHOICE Oct 2008)

"This is an excellent text for reaching students of diverse backgrounds and majors, as well as scientists from other disciplines (physics, economics, finance, and engineering) who want an introduction to functional analysis." (MAA Reviews Oct 2008)

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"A First Course in Functional Analysis is an ideal text for upper-undergraduate  and graduate-level courses in pure and applied mathematics, statistics, and engineering.  It also serves as a valuable reference for practioners across various disciplines, including the physical sciences, economics, and finance, who would like to expand their knowledge of functional analysis." (Mathematical Reviews 2009)

"It is written in a very open, nontelegraphic style, and takes care to explain topics as they come up.  Recommended." (CHOICE Oct 2008)

"This is an excellent text for reaching students of diverse backgrounds and majors, as well as scientists from other disciplines (physics, economics, finance, and engineering) who want an introduction to functional analysis." (MAA Reviews Oct 2008)

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