Linear Functional Analysis

Description

This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalized to infinite-dimensional spaces. Aimed at advanced undergraduates in mathematics and physics, the book assumes a standard background of linear algebra, real analysis (including the theory of metric spaces), and Lebesgue integration, although an introductory chapter summarizes the requisite material. The initial chapters develop the theory of infinite-dimensional normed spaces, in particular Hilbert spaces, after which the emphasis shifts to studying operators between such spaces. Functional analysis has applications to a vast range of areas of mathematics; the final chapters discuss the particularly important areas of integral and differential equations. Further highlights of the second edition include: a new chapter on the Hahn–Banach theorem and its applications to the theory of duality. This chapter also introduces the basic properties of projection operators on Banach spaces, and weak convergence of sequences in Banach spaces - topics that have applications to both linear and nonlinear functional analysis; extended coverage of the uniform boundedness theorem; plenty of exercises, with solutions provided at the back of the book. Preface......Page 6 Contents......Page 10 1. Preliminaries......Page 12 1.1 Linear Algebra......Page 13 1.2 Metric Spaces......Page 22 1.3 Lebesgue Integration......Page 31 2.1 Examples of Normed Spaces......Page 42 2.2 Finite-dimensional Normed Spaces......Page 50 2.3 Banach Spaces......Page 56 3.1 Inner Products......Page 62 3.2 Orthogonality......Page 71 3.3 Orthogonal Complements......Page 76 3.4 Orthonormal Bases in Infinite Dimensions......Page 83 3.5 Fourier Series......Page 93 4.1 Continuous Linear Transformations......Page 98 4.2 The Norm of a Bounded Linear Operator......Page 107 4.3 The Space B(X, Y )......Page 115 4.4 Inverses of Operators......Page 119 5.1 Dual Spaces......Page 132 5.2 Sublinear Functionals, Seminorms and the Hahn?Banach Theorem......Page 138 5.3 The Hahn?Banach Theorem in Normed Spaces......Page 143 5.4 The General Hahn?Banach theorem......Page 148 5.5 The Second Dual, Reflexive Spaces and Dual Operators......Page 155 5.6 Projections and Complementary Subspaces......Page 166 5.7 Weak and Weak-* Convergence......Page 170 6.1 The Adjoint of an Operator......Page 178 6.2 Normal, Self-adjoint and Unitary Operators......Page 187 6.3 The Spectrum of an Operator......Page 194 6.4 Positive Operators and Projections......Page 203 7.1 Compact Operators......Page 216 7.2 Spectral Theory of Compact Operators......Page 227 7.3 Self-adjoint Compact Operators......Page 237 8.1 Fredholm Integral Equations......Page 246 8.2 Volterra Integral Equations......Page 256 8.3 Differential Equations......Page 258 8.4 Eigenvalue Problems and Green?s Functions......Page 264 9. Solutions to Exercises......Page 276 Further Reading......Page 326 References......Page 328 Notation Index......Page 330 Index......Page 332