Functional analysis is the branch of mathematics dealing with vector spaces (with defined limit processes) and the linear operators compatible within those limits. Modern functional analysis is the study of vector spaces endowed with a topology; in particular infinite dimensional spaces. An important part of functional analysis is the extension of measure theory, integration, and probability to infinite dimensional spaces.

A course in functional analysis will most likely include many of the following topics:

  • Topological vector spaces
  • Completeness
  • Convexity
  • Duality in Banach spaces
  • Test functions and distributions
  • Fourier transforms
  • Applications to differential equations
  • Tauberian theory
  • Banach algebras
  • Bounded operators on a Hillbert space
  • Unbounded operators
  • Compactness and continuity


The Journal of Functional Analysis should be read by any student interested in this area of mathematics. Good books on functional analysis can also be found on Amazon.com and Google.

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