# 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 ${c}_{n}=\u27e8x,{e}_{n}\u27e9$.

Title | Riesz-Fischer theorem^{} |
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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 |