# Bessel inequality

Let $\mathscr{H}$ be a Hilbert space^{}, and suppose ${e}_{1},{e}_{2},\mathrm{\dots}\in \mathscr{H}$ is an orthonormal sequence. Then for any $x\in \mathscr{H}$,

$$\sum _{k=1}^{\mathrm{\infty}}{\left|\u27e8x,{e}_{k}\u27e9\right|}^{2}\le {\parallel x\parallel}^{2}.$$ |

Bessel’s inequality^{} immediately lets us define the sum

$${x}^{\prime}=\sum _{k=1}^{\mathrm{\infty}}\u27e8x,{e}_{k}\u27e9{e}_{k}.$$ |

The inequality means that the series converges.

For a complete^{} orthonormal series, we have Parseval’s theorem, which replaces inequality with equality (and consequently ${x}^{\prime}$ with $x$).

Title | Bessel inequality |
---|---|

Canonical name | BesselInequality |

Date of creation | 2013-03-22 12:46:38 |

Last modified on | 2013-03-22 12:46:38 |

Owner | ariels (338) |

Last modified by | ariels (338) |

Numerical id | 5 |

Author | ariels (338) |

Entry type | Theorem |

Classification | msc 46C05 |