Riesz-Fischer theorem

Let $\{e_n\}$ be an orthonormal basis for a (real or complex) infinite-dimensional Hilbert space $\mathscr{H}$. If $\{c_n\}$ is a sequence of (real or complex) numbers such that $\sum |c_n{|}^2$ converges, then there is an $x\in \mathscr{H}$ such that $x={\sum }_{n=1}^{\mathrm{\infty }}{c}_n{e}_n$, and $c_n=⟨x,e_n⟩$.

Title	Riesz-Fischer theorem
Canonical name	RieszFischerTheorem
Date of creation	2013-03-22 14:09:46
Last modified on	2013-03-22 14:09:46
Owner	azdbacks4234 (14155)
Last modified by	azdbacks4234 (14155)
Numerical id	7
Author	azdbacks4234 (14155)
Entry type	Theorem
Classification	msc 46C99
Related topic	LpSpace
Related topic	L2SpacesAreHilbertSpaces
Related topic	HilbertSpace
Related topic	EllpXSpace
Related topic	ClassificationOfHilbertSpaces