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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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.

Leseprobe

Preface. Acknowledgments. 1. Linear spaces and operators. 1.1 Introduction. 1.2 Linear spaces. 1.3 Linear operators. 1.4 The passage from finite-to infinite-dimensional spaces.. Exercises. 2. Normed linear spaces - the basics. 2.1 Metric spaces. 2.2 Norms. 2.3 The 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. The major Banach space theorems. 3.1 Introduction. 3.2 The Baire category theorem. 3.3 Open mappings. 3.4 Bounded inverses. 3.5 Closed linear operators. 3.6 The uniform boundedness principle. Exercises. 4. Hilbert spaces. 4.1 Introduction. 4.2 Semi-inner products. 4.3 Nearest points to convex sets. 4.4 Orthogonality. 4.5 Linear functionals on Hilbert spaces. 4.6 Linear operators on Hilbert spaces. 4.7 The order relation on the self-adjoint operators. Exercises. 5. The Hahn-Banach theorem. 5.1 Introduction. 5.2 The basic version of the Hahn-Banach theorem. 5.3 A complex version of the Hahn-Banach theorem. 5.4 Application to normed linear spaces. 5.5 Geometric versions of the 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 A review of general topology. 7.2 Topologies on linear spaces. 7.3 Linear functionals on a topological linear space. 7.4 The weak topology. 7.5 The weak topology. 7.6 Extreme points and the Krein-Milman theorem. Exercises. 8. The spectrum. 8.1 Introduction. 8.2 Banach algebras. 8.3 General properties of the spectrum. 8.4 Numerical range. 8.5 The spectrum of normal operators. 8.6 Functions of operators. 8.7 A 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 The spectrum of a compact operator. 9.5 Compact self-adjoint operators on Hilbert space. 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 The second order linear differential equation. 10.5 Sturm-Liouville problems. 10.6 The first order differential equation. 11 The spectral theorem for a bounded self-adjoint operator. 11.1 Introduction and motivation. 11.2 Spectral decomposition. 11.3 The extension of the functional calculus. 11.4 Multiplication operators. Exercises. Appendix A Zorn's lemma. Appendix B. The Stone-Weierstrass theorem. B.1 The basic theorem. B.2 Non-unital algebras. B.3 Complex algebras. Appendix C. The extended real number system and limit points of sequences. C.1 The extended reals. C.2 Limit points of sequences. Appendix D. Measure and integration. D.1 Introduction and motivation. D.2 Basic properties of measures. D.3 Properties of measurable functions. D.4 The integral of a nonnegative function. D.5 The integral of a real-valued function. D.6 The integral of a complex-valued function. D.7 Construction of Lebesgue measure on R. D.8 Competeness of measu ...