Functional Analysis by Bachman, George -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Contents Preface CHAPTER 1. Introduction to Inner Product Spaces

1.1 Some Prerequisite Material and Conventions 1.2 Inner Product Spaces 1.3 Linear Functionals, the Riesz Representation Theorem, and Adjoints Exercises 1 References

CHAPTER 2. Orthogonal Projections and the Spectral Theorem for Normal Transformations

2.1 The Complexification 2.2 Orthogonal Projections and Orthogonal Direct Sums 2.3 Unitary and Orthogonal Transformations Exercises 2 References

CHAPTER 3. Normed Spaces and Metric Spaces

3.1 Norms and Normed Linear Spaces 3.2 Metrics and Metric Spaces 3.3 Topological Notions in Metric Spaces 3.4 Closed and Open Sets, Continuity, and Homeomorphisms Exercises 3 Reference

CHAPTER 4. Isometries and Completion of a Metric Space

4.1 Isometries and Homeomorphisms 4.2 Cauchy Sequences and Complete Metric Spaces Exercises 4 Reference

CHAPTER 5. Compactness in Metric Spaces

5.1 Nested Sequences and Complete Spaces 5.2 Relative Compactness ε-Nets and Totally Bounded Sets 5.3 Countable Compactness and Sequential Compactness Exercises 5 References

CHAPTER 6. Category and Separable Spaces

6.1 Fσ Sets and Gδ Sets 6.2 Nowhere-Dense Sets and Category 6.3 The Existence of Functions Continuous Everywhere, Differentiable Nowhere 6.4 Separable Spaces Exercises 6 References

CHAPTER 7. Topological Spaces

7.1 Definitions and Examples 7.2 Bases 7.3 Weak Topologies 7.4 Separation 7.5 Compactness Exercises 7 References

CHAPTER 8. Banach Spaces, Equivalent Norms, and Factor Spaces

8.1 The Hölder and Minkowski Inequalities 8.2 Banach Spaces and Examples 8.3 The Completion of a Normed Linear Space 8.4 Generated Subspaces and Closed Subspaces 8.5 Equivalent Norms and a Theorem of Riesz 8.6 Factor Spaces 8.7 Completeness in the Factor Space 8.8 Convexity Exercises 8 References

CHAPTER 9. Commutative Convergence, Hilbert Spaces, and Bessel’s Inequality

9.1 Commutative Convergence 9.2 Norms and Inner Products on Cartesian Products of Normed and Inner Product Spaces 9.3 Hilbert Spaces 9.4 A Nonseparable Hilbert Space 9.5 Bessel’s Inequality 9.6 Some Results from L2(0, 2π) and the Riesz-Fischer Theorem 9.7 Complete Orthonormal Sets 9.8 Complete Orthonormal Sets and Parseval’s Identity 9.9 A Complete Orthonormal Set for L2(0, 2π) Appendix 9 Exercises 9 References

CHAPTER 10. Complete Orthonormal Sets

10.1 Complete Orthonormal Sets and Parseval’s Identity 10.2 The Cardinality of Complete Orthonormal Sets 10.3 A Note on the Structure of Hilbert Spaces 10.4 Closed Subspaces and the Projection Theorem for Hilbert Spaces Exercises 10 References

CHAPTER 11. The Hahn-Banach Theorem

11.1 The Hahn-Banach Theorem 11.2 Bounded Linear Functionals 11.3 The Conjugate Space Exercises 11 Appendix 11. The Problem of Measure and the Hahn-Banach Theorem Exercises 11 Appendix References

CHAPTER 12. Consequences of the Hahn-Banach Theorem

12.1 Some Consequences of the Hahn-Banach Theorem 12.2 The Second Conjugate Space 12.3 The Conjugate Space of lp 12.4 The Riesz Representation Theorem for Linear Functionals on a Hilbert Space 12.5 Reflexivity of Hilbert Spaces Exercises 12 References

CHAPTER 13. The Conjugate Space of C[a, b]

13.1 A Representation Theorem for Bounded Linear Functionals on C[a, b] 13.2 A List of Some Spaces and Their Conjugate Spaces Exercises 13 References

CHAPTER 14. Weak Convergence and Bounded Linear Transformations

14.1 Weak Convergence 14.2 Bounded Linear Transformations Exercises 14 References

CHAPTER 15. Convergence in L(X, Y) and the Principle of Uniform Boundedness

15.1 Convergence in L(X, Y) 15.2 The Principle of Uniform Boundedness 15.3 Some Consequences of the Principle of Uniform Boundedness Exercises 15 References

CHAPTER 16. Closed Transformations and the Closed Graph Theorem

16.1 The Graph of a Mapping 16.2 Closed Linear Transformations and the Bounded Inverse Theorem 16.3 Some Consequences of the Bounded Inverse Theorem Appendix 16. Supplement to Theorem 16.5 Exercises 16 References

CHAPTER 17. Closures, Conjugate Transformations, and Complete Continuity

17.1 The Closure of a Linear Transformation 17.2 A Class of Linear Transformations that Admit a Closure 17.3 The Conjugate Map of a Bounded Linear Transformation 17.4 Annihilators 17.5 Completely Continuous Operators; Finite-Dimensional Operators 17.6 Further Properties of Completely Continuous Transformations Exercises 17 References

CHAPTER 18. Spectral Notions

18.1 Spectra and the Resolvent Set 18.2 The Spectra of Two Particular Transformations 18.3 Approximate Proper Values Exercises 18 References

CHAPTER 19. Introduction to Banach Algebras

19.1 Analytic Vector-Valued Functions 19.2 Normed and Banach Algebras 19.3 Banach Algebras with Identity 19.4 An Analytic Function — the Resolvent Operator 19.5 Spectral Radius and the Spectral Mapping Theorem for Polynomials 19.6 The Gelfand Theory 19.7 Weak Topologies and the Gelfand Topology 19.8 Topological Vector Spaces and Operator Topologies Exercises 19 References

CHAPTER 20. Adjoints and Sesquilinear Functionals

20.1 The Adjoint Operator 20.2 Adjoints and Closures 20.3 Adjoints of Bounded Linear Transformations in Hilbert Spaces 20.4 Sesquilinear Functionals Exercises 20 References

CHAPTER 21. Some Spectral Results for Normal and Completely Continuous Operators

21.1 A New Expression for the Norm of A ∈ L(X, X) 21.2 Normal Transformations 21.3 Some Spectral Results for Completely Continuous Operators 21.4 Numerical Range Exercises 21 Appendix to Chapter 21. The Fredholm Alternative Theorem and the Spectrum of a Completely Continuous Transformation

A.1 Motivation A.2 The Fredholm Alternative Theorem

References

CHAPTER 22. Orthogonal Projections and Positive Definite Operators

22.1 Properties of Orthogonal Projections 22.2 Products of Projections 22.3 Positive Operators 22.4 Sums and Differences of Orthogonal Projections 22.5 The Product of Positive Operators Exercises 22 References

CHAPTER 23. Square Roots and a Spectral Decomposition Theorem

23.1 Square Root of Positive Operators 23.2 Spectral Theorem for Bounded, Normal, Finite-Dimensional Operators Exercises 23 References

CHAPTER 24. Spectral Theorem for Completely Continuous Normal Operators

24.1 Infinite Orthogonal Direct Sums: Infinite Series of Transformations 24.2 Spectral Decomposition Theorem for Completely Continuous Normal Operators Exercises 24 References

CHAPTER 25. Spectral Theorem for Bounded, Self-Adjoint Operators

25.1 A Special Case — the Self-Adjoint, Completely Continuous Operator 25.2 Further Properties of the Spectrum of Bounded, Self-Adjoint Transformations 25.3 Spectral Theorem for Bounded, Self-Adjoint Operators Exercises 25 References

CHAPTER 26. A Second Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators

26.1 A Second Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators Exercises 26 References

CHAPTER 27. A Third Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators and Some Consequences

27.1 A Third Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators 27.2 Two Consequences of the Spectral Theorem Exercises 27 References

CHAPTER 28. Spectral Theorem for Bounded, Normal Operators

28.1 The Spectral Theorem for Bounded, Normal Operators on a Hilbert Space 28.2 Spectral Measures; Unitary Transformations Exercises 28 References

CHAPTER 29. Spectral Theorem for Unbounded, Self-Adjoint Operators

29.1 Permutativity 29.2 The Spectral Theorem for Unbounded, Self-Adjoint Operators 29.3 A Proof of the Spectral Theorem Using the Cayley Transform 29.4 A Note on the Spectral Theorem for Unbounded Normal Operators Exercises 29 References

Bibliography Index of Symbols Subject Index Errata

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