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A First Course in Functional Analysis

S. David Promislow
John Wiley
Publication Date: 
Number of Pages: 
Pure and Applied Mathematics
[Reviewed by
Mihaela Poplicher
, on

This book was written by S. David Promislow, Professor Emeritus of Mathematics at York University in Toronto, Canada, as a text-book for a one-semester introductory course in functional analysis. It is based on the author’s experience in teaching such a course for upper-undergraduate and graduate students with very diverse backgrounds in mathematics, statistics, and engineering.

The prerequisites needed for reading/using the book are a knowledge of basic real analysis (sequences and series, continuity, uniform convergence of functions, topology of the real line, elements of metric spaces), as well as elementary linear algebra (linear independence, bases, matrix manipulation).

The book treats the most important topics in a first functional analysis course: linear spaces and operators, normed linear spaces, major Banach space theorems, Hilbert spaces, Hahn-Banach theorem, duality, topological linear spaces, compact operators. There are also chapters about the spectrum, applications to integral and differential equations, spectral theorem for bounded self-adjoint operators.

Most of the topics included in the book do not require knowledge beyond the basics of real analysis and linear algebra, but for people who want to study more, or to see the proofs of some of the results used and not proven (in the main chapters), there are appendices on measure and integration, Zorn’s lemma, the Stone-Weierstrass theorem, and Tychonoff’s theorem. A good list of references helps the readers who want to further their studies even deeper.

What sets this book apart is the way the proofs are presented, outlining the logic behind the steps, explaining the development of the arguments and discussing the connections between the concepts. Each section concludes with a set of exercises of different levels of difficulty which are very helpful for readers in their quest for understanding of the material.

In short, I think this is an excellent text for reaching students of diverse backgrounds and majors, as well as scientists from other disciplines (physics, economics, finance, and engineering) who want an introduction in functional analysis.

Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is



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..


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.


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.


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.


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.


6. Duality.

6.1 Examples of dual spaces.

6.2 Adjoints.

6.3 Double duals and reflexivity.

6.4 Weak and weak convergence.


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.


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.


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.


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.


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.

C.1 The extended reals.

C.2 Limit points of sequences.

Appendix D. Measure and integration.

D.1 Introduction and motivation.

D.2 Basic properties of measures.

D.3 Properties of measurable functions.

D.4 The integral of a nonnegative function.

D.5 The integral of a real-valued function.

D.6 The integral of a complex-valued function.

D.7 Construction of Lebesgue measure on R.

D.8 Competeness of measures.

D.9 Signed and complex measures.

D.10 The Radon-Nikodym derivative.

D.11 Product measures.

D.12 The Riesz representation theorem.

Appendix E. Tychono's theorem.


List of Symbols.