Short description This straight-forward, concise book is made up of carefully selected topics and is written in an accessible that requires minimal background knowledge. It provides the reader with a sense of unity in the subject's development and fully explains the essential concepts, outlining the logic behind the steps to familiarize the reader with the theories. All the important topics are covered, from linear spaces and topological vector spaces, to the main theorems such as the Major Banach Space Theorems and the Hahn-Banach Theorem. Numerous exercises of ranging difficulty are included throughout the text to enhance the reader's learning.
From the contents 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.
