An Introduction to Functional Analysis

✍ Scribed by James C. Robinson

• Publisher: Cambridge University Press
• Year: 2020
• Tongue: English
• Leaves: 422
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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 Lsup(p)
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 Lsup(p)
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