A First Course in Functional AnalysisISBN: 9780470146194
308 pages
April 2008

Description
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 classtested 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 StoneWeierstrass 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 upperundergraduate and graduatelevel 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.
Table of Contents
1. Linear Spaces and Operators.
1.1 Introduction.
1.2 Linear Spaces.
1.3 Linear Operators.
1.4 Passage from Finite to InfiniteDimensional 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 FiniteDimensional Normed Linear Spaces.
2.9 L^{p} 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 SemiInner 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 SelfAdjoint 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 CAlgebras.
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 SelfAdjoint 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 SecondOrder Linear Differential Equations.
10.5 Sturm–Liouville Problems.
10.6 FirstOrder Differential Equations.
Exercises.
11. Spectral Theorem for Bounded, SelfAdjoint 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 RealValued Function.
D.6 Integral of a ComplexValued 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.
Author Information
The Wiley Advantage
 Presents a unique coverage of the essential topics of functional analysis, based on successful lecture notes used by the author, for a concise onesemester 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.
Reviews
"A First Course in Functional Analysis is an ideal text for upperundergraduate and graduatelevel 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)
Professor Reviews
"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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