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Linear Functional Analysis

Bryan P. Rynne and Martin A. Youngson
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Allen Stenger
, on

This is an undergraduate introduction to functional analysis, with minimal prerequisites, namely linear algebra and some real analysis. The book is kept at an undergraduate level by avoiding topology (nothing but metric spaces and convergence of sequences) and by using Riemann integration and continuous functions (Lebesgue integration keeps sneaking in and being shooed away).

The book proceeds at a leisurely pace, and the reasoning is easy to follow. It is extensively cross-referenced, has a good index, a separate index of symbols (Very Good Feature), and complete solutions to all the exercises. It has numerous examples, and is especially good in giving both examples of objects that have a given property and objects that do not have the property.

It takes quite a while to get to what most people would consider functional analysis — functionals do not appear until p. 105, after a lot of material on Banach and Hilbert spaces. Topics include the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and a whole chapter on the Hahn-Banach theorem (several variants). For some reason there's almost no mention of eigenfunction expansions. There are two chapters that deal primarily with spectral theory, but the term "spectral theorem" is never mentioned, even though this theorem is stated and proved for compact self-adjoint operators (Theorem 7.34). There's little discussion of duality or weak topologies and no discussion of unbounded operators.

My big gripe with this book is that it presents functional analysis in isolation, as an abstract subject not related to any other part of mathematics or to the problems that inspired it. The bulk of the book is a sequence of Definition-Lemma-Theorem that defines some unmotivated abstract concepts and develops their properties. The last chapter tries to make up for this by looking at applications to Fredholm and Volterra integral equations and to differential equations. I think the book would be greatly improved by breaking up the material in the last chapter, and moving the problem statements to the very beginning and putting the other parts wherever there is enough background to tackle them.

A competing book is Saxe's Beginning Functional Analysis , which has the same coverage and same audience. I browsed through Saxe, and I think it does a better job of putting the subject in context, both by moving the applications earlier in the narrative and by providing brief biographies of the big names in functional analysis and what they contributed.

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.

1. Preliminaries

1.1 Linear Algebra
1.2 Metric Spaces
1.3 Lebesgue Integration

2. Normed Spaces

2.1 Examples of Normed Spaces
2.2 Finite-dimensional Normed Spaces
2.3 Banach Spaces

3. Inner Product Spaces, Hilbert Spaces

3.1 Inner Products
3.2 Orthogonality
3.3 Orthogonal Complements
3.4 Orthonormal Bases in Infinite Dimensions
3.5 Fourier Series

4. Linear Operators

4.1 Continuous Linear Transformations
4.2 The Norm of a Bounded Linear Operator
4.3 The Space B(X, Y)
4.4 Inverses of Operators

5. Duality and the Hahn-Banach Theorem

5.1 Dual Spaces
5.2 Sublinear Functionals, Seminorms and the Hahn-Banach Theorem
5.3 Hahn-Banach Theorem in Normed Spaces

5.4 The General Hahn-Banach theorem
5.5 The Second Dual, Reflexive Spaces and Dual Operators
5.6 Projections and Complementary Subspaces
5.7 Weak and Weak-* Convergence

6. Linear Operators on Hilbert Spaces

6.1 The Adjoint of an Operator
6.2 Normal, Self-adjoint and Unitary Operators
6.3 The Spectrum of an Operator
6.4 Positive Operators and Projections

7. Compact Operators

7.1 Compact Operators
7.2 Spectral Theory of Compact Operators
7.3 Self-adjoint Compact Operators

8. Integral and Differential Equations

8.1 Fredholm Integral Equations
8.2 Volterra Integral Equations
8.3 Differential Equations
8.4 Eigenvalue Problems and Green's Functions

9. Solutions to Exercises

Further Reading


Notation Index