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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 Johan.
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See Also: Riesz representation theorem

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Cross-references: converges, sequence, Hilbert space, infinite-dimensional, complex, real, orthonormal basis
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This is version 2 of Riesz-Fischer theorem, born on 2004-02-16, modified 2004-02-26.
Object id is 5586, canonical name is RieszFischerTheorem.
Accessed 3119 times total.

Classification:
AMS MSC46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous)

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another theorem? by scineram on 2006-02-25 13:09:43
As I know the Riesz-Fisher theorem states that the L^p spaces are complete. Do you know it?
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