PlanetMath: Riesz-Fischer theorem

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Riesz-Fischer theorem

(Theorem)

Let $\{e_n\}$ be an orthonormal basis for a (real or complex) infinite-dimensional Hilbert space $\mathcal{H}$ . If $\{c_n\}$ is a sequence of (real or complex) numbers such that $\sum \lvert c_n \lvert^2$ converges, then there is an $x\in \mathcal{H}$ such that $x=\sum_{n=1}^{\infty} c_n e_n$ , and $c_n = \langle x,e_n \rangle$ .

"Riesz-Fischer theorem" is owned by azdbacks4234. [ full author list (2) | owner history (2) ]

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Keywords:

Fourier coefficient, little ell, Hilbert space, orthonormal basis

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Cross-references: converges, numbers, sequence, Hilbert space, infinite-dimensional, complex, real, orthonormal basis
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This is version 4 of Riesz-Fischer theorem, born on 2004-02-16, modified 2008-11-08.
Object id is 5586, canonical name is RieszFischerTheorem.
Accessed 4262 times total.

Classification:

46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous)

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