# Orthogonal basis

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A system of pairwise orthogonal non-zero elements of a Hilbert space , such that any element can be (uniquely) represented in the form of a norm-convergent series called the Fourier series of the element with respect to the system . The basis is usually chosen such that , and is then called an orthonormal basis. In this case, the numbers , called the Fourier coefficients of the element relative to the orthonormal basis , take the form . A necessary and sufficient condition for an orthonormal system to be a basis is the Parseval–Steklov equality for any . A Hilbert space which has an orthonormal basis is separable and, conversely, in any separable Hilbert space an orthonormal basis exists. If an arbitrary system of numbers is given such that , then in the case of a Hilbert space with a basis , the series converges in norm to an element . An isomorphism between any separable Hilbert space and the space is established in this way (Riesz–Fischer theorem).

How to Cite This Entry:
Orthogonal basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_basis&oldid=14572
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article