فضاء هلبرت

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

• Harmonic analysis
• Hermitian operators
• Hilbert C*-module
• Hilbert manifold
• Mathematical analysis
• Operator algebra
• Rigged Hilbert space
• Reproducing kernel Hilbert space
• Topologies on the set of operators on a Hilbert space

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