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A Friendly Approach to Functional Analysis

Amol Sasane
World Scientific
Publication Date: 
Number of Pages: 
Essential Textbooks in Mathematics
[Reviewed by
Allen Stenger
, on

This is a very gradual introduction to functional analysis, aimed at undergraduates. It takes an expansive view of functional analysis, defining it as “calculus in the setting of vector spaces” (p. vii). By this it means calculus of real-valued functions defined on normed linear spaces. The book does not assume the Lebesgue integral, and doesn’t tackle integration of functions on these spaces, so calculus here means mostly convergence of sequences and differentiation, with no integration except on the real line. The book is unusual among functional analysis books in devoting a lot of space to the derivative.

The “friendly” aspect promised in the title is not explained, but there are three things I think would strike most students as friendly: the slow pace, the enormous number of examples, and complete solutions to all the exercises. The prerequisites are minimal: some calculus and some linear algebra. The book develops all the topology needed from scratch, and spends most of its time developing properties of normed linear spaces and inner product spaces. It hedges on the Lebesgue integral: it does use it for the real line throughout, but in a way that does not require any detailed knowledge, and it advises the student to think of it as the Riemann integral. There’s a brief appendix outlining properties of the Lebesgue integral on the real line.

The book includes a good assortment of applications, mostly in physics. These include the Euler–Lagrange equation in the calculus of variations (with examples), the wave equation, Hamiltonian mechanics, the isoperimetric theorem, and a brief survey of quantum mechanics including the uncertainty principle. This list is a little misleading, in that the applications (except the quantum mechanics) are treated in a very classical way with partial derivatives and Fourier series. They don’t really draw on functional analysis except perhaps in viewpoint (we seek members of a function space having certain properties).

The book has about 200 exercises (all with detailed solutions). They range from easy to moderately difficult, and are mostly to work out properties of particular mappings or subspaces, along with a few that extend results in the text or tie up loose ends.

This may be that rare thing: a book with too many examples. I thought they often interfered with the exposition and made it hard to see where we were going. It does manage to fit in all the classically-important theorems of functional analysis (Open Mapping, Theorem, Hahn–Banach Theorem, Banach–Steinhaus theorem, Riesz Representation Theorem, spectral theorem) but they were swamped by the examples. It doesn’t deal explicitly with the Riesz–Fischer theorem, but it does define orthonormal bases and gives several examples (with proofs). A more compact book, with the same prerequisites but maybe less friendly, is Saxe’s Beginning Functional Analysis. It covers more topics, and in particular develops the theory of Lebesgue integration on general spaces.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

  • Normed and Banach Spaces
  • Continuous and Linear Maps
  • Differentiation
  • Geometry of Inner Product Spaces
  • Compact Operators
  • A Glimpse of Distribution Theory