A First Course in Functional Analysis

1. Linear Spaces and Operators.

1.1 Introduction.

1.2 Linear Spaces.

1.3 Linear Operators.

1.4 Passage from Finite- to Infinite-Dimensional Spaces.

Exercises.

2. Normed Linear Spaces: The Basics.

2.1 Metric Spaces.

2.2 Norms.

2.3 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 L^p Spaces.

2.10 Direct Products and Sums.

2.11 Schauder Bases.

2.12 Fixed Points and Contraction Mappings.

Exercises.

3. Major Banach Space Theorems.

3.1 Introduction.

3.2 Baire Category Theorem.

3.3 Open Mappings.

3.4 Bounded Inverses.

3.5 Closed Linear Operators.

3.6 Uniform Boundedness Principle.

Exercises.

4. Hilbert Spaces.

4.1 Introduction.

4.2 Semi-Inner Products.

4.3 Nearest Points and Convexity.

4.4 Orthogonality.

4.5 Linear Functionals on Hilbert Spaces.

4.6 Linear Operators on Hilbert Spaces.

4.7 Order Relation on Self-Adjoint Operators.

Exercises.

5. Hahn–Banach Theorem.

5.1 Introduction.

5.2 Basic Version of Hahn–Banach Theorem.

5.3 Complex Version of Hahn–Banach Theorem.

5.4 Application to Normed Linear Spaces.

5.5 Geometric Versions of 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 Review of General Topology.

7.2 Topologies on Linear Spaces.

7.3 Linear Functionals on Topological Linear Spaces.

7.4 Weak Topology.

7.5 Weak* Topology.

7.6 Extreme Points and Krein–Milman Theorem.

7.7 Operator Topologies.

Exercises.

8. The Spectrum.

8.1 Introduction.

8.2 Banach Algebras.

8.3 General Properties of the Spectrum.

8.4 Numerical Range.

8.5 Spectrum of a Normal Operator.

8.6 Functions of Operators.

8.7 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 Spectrum of a Compact Operator.

9.5 Compact Self-Adjoint Operators on Hilbert Spaces.

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 Second-Order Linear Differential Equations.

10.5 Sturm–Liouville Problems.

10.6 First-Order Differential Equations.

Exercises.

11. Spectral Theorem for Bounded, Self-Adjoint Operators.

11.1 Introduction and Motivation.

11.2 Spectral Decomposition.

11.3 Extension of Functional Calculus.

11.4 Multiplication Operators.

Exercises.

Appendix A Zorn's Lemma.

Appendix B Stone–Weierstrass Theorem.

B.1 Basic Theorem.

B.2 Nonunital Algebras.

B.3 Complex Algebras.

Appendix C Extended Real Numbers and Limit Points of Sequences.

C.1 Extended Reals.

C.2 Limit Points of Sequences.

Appendix D Measure and Integration.

D.1 Introduction and Notation.

D.2 Basic Properties of Measures 258

D.3 Properties of Measurable Functions.

D.4 Integral of a Nonnegative Function.

D.5 Integral of an Extended Real-Valued Function.

D.6 Integral of a Complex-Valued Function.

D.7 Construction of Lebesgue Measure on R.

D.8 Completeness of Measures.

D.9 Signed and Complex Measures.

D.10 Radon–Nikodym Derivatives.

D.11 Product Measures.

D.12 Riesz Representation Theorem.

Appendix E Tychonoff's Theorem.

Symbols.

References.

Index.