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Hilbert spaces

  • Luigi Ambrosio
  • Giuseppe Da Prato
  • Andrea Mennucci
Part of the Appunti/Lecture Notes book series (PSNS, volume 10)

Abstract

In this chapter we recall the basic facts regarding real vector spaces endowed with a scalar product. We introduce the concept of Hilbert space and show that, even for the infinite-dimensional ones, continuous linear functionals are induced by the scalar product. Moreover, we see that even in some classes of infinite dimensional spaces (the so-called separable ones) there exists a well-defined notion of basis (the so-called complete orthonormal systems), obtained replacing finite sums with converging series. Even though the presentation will be self-contained, we assume that the reader has already some familiarity with these concepts (basis, scalar product, representation of linear functionals) in finite-dimensional spaces.

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Copyright information

© Scuola Normale Superiore Pisa 2011

Authors and Affiliations

  • Luigi Ambrosio
    • 1
  • Giuseppe Da Prato
    • 1
  • Andrea Mennucci
    • 1
  1. 1.Scuola Normale SuperiorePisaItaly

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