First Course in Functional Analysis

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.

"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)

• Preface xi1. Linear Spaces and Operators 11.1 Introduction 11.2 Linear Spaces 21.3 Linear Operators 51.4 Passage from Finite- to Infinite-Dimensional Spaces 7Exercises 82. Normed Linear Spaces: The Basics 112.1 Metric Spaces 112.2 Norms 122.3 Space of Bounded Functions 182.4 Bounded Linear Operators 192.5 Completeness 212.6 Comparison of Norms 302.7 Quotient Spaces 312.8 Finite-Dimensional Normed Linear Spaces 342.9 Lᵖ Spaces 382.10 Direct Products and Sums 512.11 Schauder Bases 532.12 Fixed Points and Contraction Mappings 53Exercises 543. Major Banach Space Theorems 593.1 Introduction 593.2 Baire Category Theorem 593.3 Open Mappings 613.4 Bounded Inverses 633.5 Closed Linear Operators 643.6 Uniform Boundedness Principle 66Exercises 684. Hilbert Spaces 714.1 Introduction 714.2 Semi-Inner Products 724.3 Nearest Points and Convexity 774.4 Orthogonality 804.5 Linear Functionals on Hilbert Spaces 864.6 Linear Operators on Hilbert Spaces 884.7 Order Relation on Self-Adjoint Operators 97Exercises 985. Hahn–Banach Theorem 1035.1 Introduction 1035.2 Basic Version of Hahn–Banach Theorem 1045.3 Complex Version of Hahn–Banach Theorem 1055.4 Application to Normed Linear Spaces 1075.5 Geometric Versions of Hahn–Banach Theorem 108Exercises 1186. Duality 1216.1 Examples of Dual Spaces 1216.2 Adjoints 1306.3 Double Duals and Reflexivity 1336.4 Weak and Weak* Convergence 136Exercises 1407. Topological Linear Spaces 1437.1 Review of General Topology 1437.2 Topologies on Linear Spaces 1487.3 Linear Functionals on Topological Linear Spaces 1517.4 Weak Topology 1537.5 Weak* Topology 1567.6 Extreme Points and Krein–Milman Theorem 1607.7 Operator Topologies 164Exercises 1648. The Spectrum 1678.1 Introduction 1678.2 Banach Algebras 1698.3 General Properties of the Spectrum 1708.4 Numerical Range 1768.5 Spectrum of a Normal Operator 1778.6 Functions of Operators 1808.7 Brief Introduction to C_-Algebras 183Exercises 1849. Compact Operators 1879.1 Introduction and Basic Definitions 1879.2 Compactness Criteria in Metric Spaces 1889.3 New Compact Operators from Old 1929.4 Spectrum of a Compact Operator 1949.5 Compact Self-Adjoint Operators on Hilbert Spaces 1979.6 Invariant Subspaces 201Exercises 20310. Application to Integral and Differential Equations 20510.1 Introduction 20510.2 Integral Operators 20610.3 Integral Equations 21110.4 Second-Order Linear Differential Equations 21410.5 Sturm–Liouville Problems 21710.6 First-Order Differential Equations 223Exercises 22611. Spectral Theorem for Bounded, Self-Adjoint Operators 22911.1 Introduction and Motivation 22911.2 Spectral Decomposition 23111.3 Extension of Functional Calculus 23511.4 Multiplication Operators 240Exercises 243Appendix A Zorn’s Lemma 245Appendix B Stone–Weierstrass Theorem 247B.1 Basic Theorem 247B.2 Nonunital Algebras 250B.3 Complex Algebras 252Appendix C Extended Real Numbers and Limit Points of Sequences 253C.1 Extended Reals 253C.2 Limit Points of Sequences 254Appendix D Measure and Integration 257D.1 Introduction and Notation 257D.2 Basic Properties of Measures 258D.3 Properties of Measurable Functions 259D.4 Integral of a Nonnegative Function 261D.5 Integral of an Extended Real-Valued Function 265D.6 Integral of a Complex-Valued Function 267D.7 Construction of Lebesgue Measure on R 267D.8 Completeness of Measures 273D.9 Signed and Complex Measures 274D.10 Radon–Nikodym Derivatives 276D.11 Product Measures 278D.12 Riesz Representation Theorem 280Appendix E Tychonoff’s Theorem 289Symbols 293References 297Index 299