# Riesz sequence

## Definition

A sequence of vectors $({x}_{n})$ in a Hilbert space^{} $H$ is called a Riesz sequence if there exist constants $$ such that

$$c\left(\sum _{n}{|{a}_{n}|}^{2}\right)\le {\parallel \sum _{n}{a}_{n}{x}_{n}\parallel}^{2}\le C\left(\sum _{n}{|{a}_{n}|}^{2}\right)$$ |

for all sequences of scalars $({a}_{n})\in {l}^{2}$. A Riesz sequence is called a Riesz basis if $\overline{span({x}_{n})}=H$.

If $H$ is a finite-dimensional space, then every basis of $H$ is a Riesz basis.

Title | Riesz sequence |
---|---|

Canonical name | RieszSequence |

Date of creation | 2013-03-22 14:26:53 |

Last modified on | 2013-03-22 14:26:53 |

Owner | swiftset (1337) |

Last modified by | swiftset (1337) |

Numerical id | 7 |

Author | swiftset (1337) |

Entry type | Definition |

Classification | msc 46C05 |

Related topic | OrthonormalBasis |

Related topic | HilbertSpace |

Related topic | Frame2 |

Related topic | AnEquivalentConditionForTheTranslatesOfAnL_2FunctionToFormARieszBasis |

Defines | Riesz basis |