An Introduction to Functional Analysis

"This accessible text covers key results in functional analysis that are essential for further study in the calculus of variations, analysis, dynamical systems, and the theory of partial differential equations. The treatment of Hilbert spaces covers the topics required to prove the Hilbert-Schmidt Theorem, including orthonormal bases; the Riesz Representation Theorem; and the basics of spectral theory. The material on Banach spaces and their duals includes the Hahn-Banach Theorem, the Krein-Milman Theorem, and results based on the Baire Category Theorem, before culminating in a proof of sequential weak compactness in reflexive spaces. Arguments are presented in detail, and more than 200 fully worked exercises are included to provide practice applying techniques and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces and point-set topology is assumed, but knowledge of measure theory is not required, making this book ideal for upper undergraduate-level and beginning graduate-level courses."--Quatrième de couverture Cover Half-title page Title page Copyright page Dedication Contents Preface Part I Preliminaries 1 Vector Spaces and Bases 1.1 Definition of a Vector Space 1.2 Examples of Vector Spaces 1.3 Linear Subspaces 1.4 Spanning Sets, Linear Independence, and Bases 1.5 Linear Maps between Vector Spaces and Their Inverses 1.6 Existence of Bases and Zorn’s Lemma Exercises 2 Metric Spaces 2.1 Metric Spaces 2.2 Open and Closed Sets 2.3 Continuity and Sequential Continuity 2.4 Interior, Closure, Density, and Separability 2.5 Compactness Exercises Part II Normed Linear Spaces 3 Norms and Normed Spaces 3.1 Norms 3.2 Examples of Normed Spaces 3.3 Convergence in Normed Spaces 3.4 Equivalent Norms 3.5 Isomorphisms between Normed Spaces 3.6 Separability of Normed Spaces Exercises 4 Complete Normed Spaces 4.1 Banach Spaces 4.2 Examples of Banach Spaces 4.2.1 Sequence Spaces 4.2.2 Spaces of Functions 4.3 Sequences in Banach Spaces 4.4 The Contraction Mapping Theorem Exercises 5 Finite-Dimensional Normed Spaces 5.1 Equivalence of Norms on Finite-Dimensional Spaces 5.2 Compactness of the Closed Unit Ball Exercises 6 Spaces of Continuous Functions 6.1 The Weierstrass Approximation Theorem 6.2 The Stone–Weierstrass Theorem 6.3 The Arzelà–Ascoli Theorem Exercises 7 Completions and the Lebesgue Spaces L[sup(p)]([Omega]) 7.1 Non-completeness of C([0, 1]) with the L[sup(1)] Norm 7.2 The Completion of a Normed Space 7.3 Definition of the L[sup(p)] Spaces as Completions Exercises Part III Hilbert Spaces 8 Hilbert Spaces 8.1 Inner Products 8.2 The Cauchy–Schwarz Inequality 8.3 Properties of the Induced Norms 8.4 Hilbert Spaces Exercises 9 Orthonormal Sets and Orthonormal Bases for Hilbert Spaces 9.1 Schauder Bases in Normed Spaces 9.2 Orthonormal Sets 9.3 Convergence of Orthogonal Series 9.4 Orthonormal Bases for Hilbert Spaces 9.5 Separable Hilbert Spaces Exercises 10 Closest Points and Approximation 10.1 Closest Points in Convex Subsets of Hilbert Spaces 10.2 Linear Subspaces and Orthogonal Complements 10.3 Best Approximations Exercises 11 Linear Maps between Normed Spaces 11.1 Bounded Linear Maps 11.2 Some Examples of Bounded Linear Maps 11.3 Completeness of B(X, Y ) When Y Is Complete 11.4 Kernel and Range 11.5 Inverses and Invertibility Exercises 12 Dual Spaces and the Riesz Representation Theorem 12.1 The Dual Space 12.2 The Riesz Representation Theorem Exercises 13 The Hilbert Adjoint of a Linear Operator 13.1 Existence of the Hilbert Adjoint 13.2 Some Examples of the Hilbert Adjoint Exercises 14 The Spectrum of a Bounded Linear Operator 14.1 The Resolvent and Spectrum 14.2 The Spectral Mapping Theorem for Polynomials Exercises 15 Compact Linear Operators 15.1 Compact Operators 15.2 Examples of Compact Operators 15.3 Two Results for Compact Operators Exercises 16 The Hilbert–Schmidt Theorem 16.1 Eigenvalues of Self-Adjoint Operators 16.2 Eigenvalues of Compact Self-Adjoint Operators 16.3 The Hilbert–Schmidt Theorem Exercises 17 Application: Sturm–Liouville Problems 17.1 Symmetry of L and the Wronskian 17.2 The Green’s Function 17.3 Eigenvalues of the Sturm–Liouville Problem Part IV Banach Spaces 18 Dual Spaces of Banach Spaces 18.1 The Young and Hölder Inequalities 18.2 The Dual Spaces of l[sup(p)] 18.3 Dual Spaces of L[sup(p)](Omega) Exercises 19 The Hahn–Banach Theorem 19.1 The Hahn–Banach Theorem: Real Case 19.2 The Hahn–Banach Theorem: Complex Case Exercises 20 Some Applications of the Hahn–Banach Theorem 20.1 Existence of a Support Functional 20.2 The Distance Functional 20.3 Separability of X∗ Implies Separability of X 20.4 Adjoints of Linear Maps between Banach Spaces 20.5 Generalised Banach Limits Exercises 21 Convex Subsets of Banach Spaces 21.1 The Minkowski Functional 21.2 Separating Convex Sets 21.3 Linear Functionals and Hyperplanes 21.4 Characterisation of Closed Convex Sets 21.5 The Convex Hull 21.6 The Krein–Milman Theorem Exercises 22 The Principle of Uniform Boundedness 22.1 The Baire Category Theorem 22.2 The Principle of Uniform Boundedness 22.3 Fourier Series of Continuous Functions Exercises 23 The Open Mapping, Inverse Mapping, and Closed Graph Theorems 23.1 The Open Mapping and Inverse Mapping Theorems 23.2 Schauder Bases in Separable Banach Spaces 23.3 The Closed Graph Theorem Exercises 24 Spectral Theory for Compact Operators 24.1 Properties of T − I When T Is Compact 24.2 Properties of Eigenvalues 25 Unbounded Operators on Hilbert Spaces 25.1 Adjoints of Unbounded Operators 25.2 Closed Operators and the Closure of Symmetric Operators 25.3 The Spectrum of Closed Unbounded Self-Adjoint Operators 26 Reflexive Spaces 26.1 The Second Dual 26.2 Some Examples of Reflexive Spaces 26.3 X Is Reflexive If and Only If X[sup(∗)] Is Reflexive Exercises 27 Weak and Weak-∗ Convergence 27.1 Weak Convergence 27.2 Examples of Weak Convergence in Various Spaces 27.2.1 Weak Convergence in l[sup(p)], 1 < p < ∞ 27.2.2 Weak Convergence in l[sup(1)]: Schur’s Theorem 27.2.3 Weak versus Pointwise Convergence in C([0, 1]) 27.3 Weak Closures 27.4 Weak-∗ Convergence 27.5 Two Weak-Compactness Theorems Exercises Appendices Appendix A Zorn’s Lemma Appendix B Lebesgue Integration Appendix C The Banach–Alaoglu Theorem Solutions to Exercises References Index "This accessible text covers key results in functional analysis that are essential for further study in the calculus of variations, analysis, dynamical systems, and the theory of partial differential equations. The treatment of Hilbert spaces covers the topics required to prove the Hilbert-Schmidt Theorem, including orthonormal bases; the Riesz Representation Theorem; and the basics of spectral theory. The material on Banach spaces and their duals includes the Hahn-Banach Theorem, the Krein-Milman Theorem, and results based on the Baire Category Theorem, before culminating in a proof of sequential weak compactness in reflexive spaces. Arguments are presented in detail, and more than 200 fully worked exercises are included to provide practice applying techniques and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces and point-set topology is assumed, but knowledge of measure theory is not required, making this book ideal for upper undergraduate-level and beginning graduate-level courses"-- Provided by publisher