# Riesz-Fischer 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 $c_{n}=\langle x,e_{n}\rangle$.

Title Riesz-Fischer theorem RieszFischerTheorem 2013-03-22 14:09:46 2013-03-22 14:09:46 azdbacks4234 (14155) azdbacks4234 (14155) 7 azdbacks4234 (14155) Theorem msc 46C99 LpSpace L2SpacesAreHilbertSpaces HilbertSpace EllpXSpace ClassificationOfHilbertSpaces