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Functional Analysis

George Bachman and Lawrence Narici
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a clearly-written, leisurely, and discursive textbook of functional analysis, slanted toward orthogonal expansions and the spectral theorem. The present volume is an unaltered reprint of the work published in 1966 by Academic Press.

The approach is concrete, with most of the theorems and definitions either being developed out of specific examples or being immediately applied to specific examples. The first few chapters prove many of the later theorems in the specific case of finite-dimensional spaces, and later chapters come back and prove the results in full generality.

The prerequisites are low and are stated to be advanced calculus and linear algebra. The first quarter of the book develops the necessary background about metric, normed, Hilbert, and topological spaces. The second quarter of the book contains the core of functional analysis, centering on the Hahn-Banach Theorem, the Uniform Boundedness Principle, and the Closed-Graph Theorem. The last half of the book is devoted to “spectral notions” and many versions of the spectral theorem. The book does not cover all topics in functional analysis, and in particular it says very little about applications, but it covers the essentials and goes into a great deal of detail on the topics it does cover.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


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

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

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

Chapter 4. Isometries and Completion of a Metric Space
4.1 Isometries and Homeomorphisms
4.2 Cauchy Sequences and Complete Metric Spaces

Chapter 5. Compactness in Metric Spaces
5.1 Nested Sequences and Complete Spaces
5.2 Relative Compactness and Sequential Compactness
5.3 Countable Compactness and Sequential Compactness

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

Chapter 7. Topological Spaces
7.1 Definitions and Examples
7.2 Bases
7.3 Weak Topologies
7.4 Separation
7.5 Compactness

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

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(O, 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(O, 2π)

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

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

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

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

CHAPTER 14. Weak Convergence and Bounded Linear Transformations
14.1 Weak Convergence
14.2 Bounded Linear Transformations

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

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

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

Chapter 18. Spectral Notions
18.1 Spectra and the Resolvent Set
18.2 The Spectra of Two Particular Transformations
18.3 Approximate Proper Values

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

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

Chapter 21. Some Spectral Results for Normal and Completely Continuous Operators
21.1 A New Expression for the Norm of A in L(X,X)
21.2 Normal Transformations
21.3 Some Spectral Results for Completely Continuous Operators
21.4 Numerical Range
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

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

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

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

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

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

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

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

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


Index of Symbols

Subject Index