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Other titles in the Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts series:
Pure and Applied Mathematics: A Wiley-Interscience Series of #86: A First Course in Functional Analysisby S. David Promislow
Synopses & Reviews
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.
Book News Annotation:
Promislow (emeritus, mathematics, York U., Canada) introduces the main ideas of functional analysis to beginning graduate or advanced undergraduate students with a prior knowledge of basic real analysis and such topics of elementary linear algebra as linear independence, bases, and matrix manipulation. He assumes no prior knowledge of measure theory or general topology. Chapters cover linear spaces and operators, normed linear spaces, major Banach space theorems, Hilbert spaces, Hahn-Banach theorem, duality, topological linear spaces, the spectrum, compact operators, application to integral and differential equations, and spectral theorem for bounded self-adjoint operators. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)
Requiring only a preliminary knowledge of elementary linear algebra and real analysis, this book provides an introduction to the basic principles and practical applications of functional analysis. Based on the author’s own class-tested material, the book uses clear language to explain the major concepts of functional analysis. 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.
This straight-forward, concise book is made up of carefully selected topics and is written in an accessible that requires minimal background knowledge. It provides the reader with a sense of unity in the subject's development and fully explains the essential concepts, outlining the logic behind the steps to familiarize the reader with the theories. All the important topics are covered, from linear spaces and topological vector spaces, to the main theorems such as the Major Banach Space Theorems and the Hahn-Banach Theorem. Numerous exercises of ranging difficulty are included throughout the text to enhance the reader's learning.
About the Author
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.
Table of Contents
1. Linear Spaces and Operators.
1.2 Linear Spaces.
1.3 Linear Operators.
1.4 Passage from Finite- to Infinite-Dimensional Spaces.
2. Normed Linear Spaces: The Basics.
2.1 Metric Spaces.
2.3 Space of Bounded Functions.
2.4 Bounded Linear Operators.
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.
3. Major Banach Space Theorems.
3.2 Baire Category Theorem.
3.3 Open Mappings.
3.4 Bounded Inverses.
3.5 Closed Linear Operators.
3.6 Uniform Boundedness Principle.
4. Hilbert Spaces.
4.2 Semi-Inner Products.
4.3 Nearest Points and Convexity.
4.5 Linear Functionals on Hilbert Spaces.
4.6 Linear Operators on Hilbert Spaces.
4.7 Order Relation on Self-Adjoint Operators.
5. Hahn–Banach Theorem.
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.
6.1 Examples of Dual Spaces.
6.3 Double Duals and Reflexivity.
6.4 Weak and Weak* Convergence.
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.
8. The Spectrum.
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.
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.
10. Application to Integral and Differential Equations.
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.
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.
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.
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