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Elementary Functional Analysis

Charles Swartz
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a very concise but comprehensive introduction to functional analysis. The prerequisites are minimal: some real analysis and linear algebra, but no topology or Lebesgue integration.

The book’s strength is the wide variety of applications of functional analysis it gives, to many diverse areas of mathematics. There is quite a lot about abstract spaces, and these are illustrated with many examples of specific function spaces, but not the Lp spaces (because the Lebesgue integral is not used). About one-quarter of the book is devoted to various forms of the spectral theorem. Apart from these important features the book is fairly conventional and covers the standard topics.

Karen Saxe’s Beginning Functional Analysis is a very different book that has the same prerequisites. Saxe’s book gives a carefully selected coverage of the most important and interesting results of functional analysis, without any attempt to be comprehensive. Swartz’s book is an impressive accomplishment, presenting quite a lot of functional analysis and its applications in only 180 pages and with almost no prerequisites. But all in all I think Saxe’s book is the better choice for a course at this level, not least because it is very charming.

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.

  • Normed Linear and Banach Spaces
  • Linear Operators
  • Quotient Spaces
  • Finite Dimensional Normed Spaces
  • Inner Product and Hilbert Spaces
  • The Hahn–Banach Theorem
  • Applications of the Hahn–Banach Theorem to Normed Spaces
  • The Uniform Boundedness Principle
  • Weak Convergence
  • The Open Mapping and Closed Graph Theorems
  • Projections
  • Schauder Basis
  • Transpose and Adjoints of Continuous Linear Operators
  • Compact Operators
  • The Fredholm Alternative
  • The Spectrum of an Operator
  • Subdivisions of the Spectrum
  • The Spectrum of a Compact Operator
  • Symmetric Linear Operators
  • The Spectral Theorem for Compact Symmetric Operators
  • Symmetric Operators with Compact Inverse
  • Bounded Self Adjoint Operators
  • Orthogonal Projections
  • Sesquilinear Functionals
  • The Spectral Theorem for Bounded Self Adjoint Operators
  • An Operational Calculus
  • The Spectral Theorem for Normal Operators