# Banach-Steinhaus theorem

Let $X$ be a Banach space^{} and $Y$ a normed space^{}.
If a family $\mathcal{F}\subset \mathcal{B}(X,Y)$ of bounded operators^{} from $X$ to $Y$ satisfies

$$ |

for each $x\in X$, then

$$ |

i.e. $\mathcal{F}$ is a bounded^{} subset of $\mathcal{B}(X,Y)$
with the usual operator norm. In other words,
there exists a constant $c$ such that for all $x\in X$ and $T\in \mathcal{F}$,

$$\parallel Tx\parallel \le c\parallel x\parallel .$$ |

Title | Banach-Steinhaus theorem |
---|---|

Canonical name | BanachSteinhausTheorem |

Date of creation | 2013-03-22 14:48:39 |

Last modified on | 2013-03-22 14:48:39 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 5 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 46B99 |

Synonym | Principle of Uniform Boundedness |

Synonym | Uniform Boundedness Principle |