Hilbert space

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Hilbert spaces can be used to study the harmonics of vibrating strings.

The mathematical concept of a Hilbert space generalizes on the idea of Euclidean space. It takes the mathematics used in two and three dimensions, and asks what happens if there are more than three dimensions. It is named after David Hilbert.

It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits in the space to let the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer). Hilbert spaces are used in ergodic theory which forms the mathematical underpinning of the study of thermodynamics. John von Neumann coined the term "Hilbert space" for the abstract concept underlying many of these diverse applications. The success of Hilbert space methods started a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

References[change | change source]

      , MR 1729490
    .
      .
      .
      .
      .
  • Brenner, S.; Scott, R. L. (2005), The Mathematical Theory of Finite Element Methods (2nd ed.), Springer, ISBN 0-3879-5451-1
      .
  • Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan (1998), One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications, 15, The Clarendon Press Oxford University Press, ISBN 978-0-19-850465-8
      , MR 1694383
    .
  • Clarkson, J. A. (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc. 40 (3): 396–414, doi:10.2307/1989630
     , JSTOR 1989630
       .
      , http://books.google.com/books?as_isbn=0821847902.
  • Folland, Gerald B. (1989), Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, ISBN 0-691-08527-7
      .
  • Fréchet, Maurice (1907), "Sur les ensembles de fonctions et les opérations linéaires", C. R. Acad. Sci. Paris 144: 1414–1416.
  • Fréchet, Maurice (1904–1907), Sur les opérations linéaires.
  • Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, World Scientific, ISBN 981-238-043-4
      .
      , MR 1807717
    .
  • Halmos, Paul (1957), Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea Pub. Co
  • Halmos, Paul (1982), A Hilbert Space Problem Book, Springer-Verlag, ISBN 0387906851
      .
     , http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D27779.
     , JSTOR 2313748
       .
      , MR 1468229
    .
    .
      .
      .
      .
      , http://books.google.com/books?as_isbn=048665656X.
  • Lindenstrauss, J.; Tzafriri, L. (1971), "On the complemented subspaces problem", Israel Journal of Mathematics 9 (2): 263–269, doi:10.1007/BF02771592
     , ISSN 0021-2172
      , MR 0276734
    .
      , http://www.encyclopediaofmath.org/index.php?title=H/h047380.
    .
  • Prugovečki, Eduard (1981), Quantum mechanics in Hilbert space (2nd ed.), Dover (published 2006), ISBN 978-0486453279
      .
      .
      .
      .
      .
      ; originally published Monografje Matematyczne, vol. 7, Warszawa, 1937.
     .
      , MR 883081
    .
  • Sobrino, Luis (1996), Elements of non-relativistic quantum mechanics, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN 9789810223861
      , MR 1626401
    .
  • Stewart, James (2006), Calculus: Concepts and Contexts (3rd ed.), Thomson/Brooks/Cole.
  • Stein, E (1970), Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, ISBN 0-691-08079-8
      .
      .
     .
         , doi:10.1073/pnas.18.3.263
     , JSTOR 86260
       , PMC 1076204
     , PMID 16587674
      .
      , MR 1435976
    .
      .
      , MR 566954
    .
      .
  • Young, Nicholas (1988), An introduction to Hilbert space, Cambridge University Press, ISBN 0-521-33071-8
      , Zbl 0645.46024
      .

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