فضاء هلبرت

(تم التحويل من Hilbert space)
Hilbert spaces can be used to study the harmonics of vibrating strings.

فضاء هيلبرت هو فضاء معياري معرف عليه دالة الضرب الداخلي بشرط أن يكون المعيار المعرف عليه هو بدلالة دالة الضرب الداخلي هذه, بالإضافة إلى وجوب كونه فضاء معياري كامل أو ما يدعى ب فضاء باناخ. وهذا يعني أن أي فضاء هيلبرت هو فضاء باناخ ولكن العكس غير صحيح.

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مقدمة وتاريخ

David Hilbert, the mathematician whose groundbreaking work on the study of integral equations and quadratic forms led to the introduction of the Hilbert space.


التطبيقات

The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy. The energy is given by the (time-independent) Schrödinger operator acting on a dense subspace of the Hilbert space of square-integrable functions on R3, and its spectrum determines the possible energy levels.


Parallelogram identity and polarization

Geometrically, the parallelogram identity asserts that AC2 + BD2 = 2(AB2 + AD2). In words, the sum of the squares of the diagonals is twice the sum of the squares of any two adjacent sides.


طالع ايضاً

هناك كتاب ، Functional Analysis/Hilbert spaces، في معرفة الكتب.


الهامش

المصادر

  • Bers, Lipman; John, Fritz; Schechter, Martin (1981), Partial differential equations, American Mathematical Society, ISBN 0821800493 .
  • Bourbaki, Nicolas (1986), Spectral theories, Elements of mathematics, Berlin: Springer-Verlag, ISBN 0201007673 
  • Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3540136279 
  • Brenner, S.; Scott, R. L. (2005), The Mathematical Theory of Finite Element Methods (2nd ed.), Springer, ISBN 0-3879-5451-1 .
  • Clarkson, J. A. (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc. 40: 396–414, http://www.jstor.org/stable/1989630 .
  • Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Vol. I, Interscience 
  • Dieudonné, Jean (1960), Foundations of Modern Analysis, Academic Press .
  • Dunford, N.; Schwartz, J.T. (1958), Linear operators, Parts I and II, Wiley-Interscience .
  • Duren, P. (1970), Theory of -Spaces, New York: Academic Press .
  • 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 linéares", C. R. Acad. Sci. Paris 144: 1414-1416 .
  • Fréchet, Maurice (1904-1907), Sur les opérations linéares .
  • Giusti, Enrico (2003), Direct Methods in the Calculus of Variations, World Scientific, ISBN 981-238-043-4 .
  • 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