# pointwise limit of bounded operators is bounded

The following result is a corollary of the principle of uniform boundedness^{}.

Theorem - Let $X$ be a Banach space^{} and $Y$ a normed vector space^{}. Let $({T}_{n})\in B(X,Y)$ be a sequence of bounded operators^{} from $X$ to $Y$. If $({T}_{n}x)$ converges for every $x\in X$, then the operator

$T:X\u27f6Y$

$$Tx=\underset{n\to \mathrm{\infty}}{lim}{T}_{n}x$$ |

is linear and . Moreover, the sequence $(\parallel {T}_{n}\parallel )$ is bounded (http://planetmath.org/Bounded).

Proof : It is clear that the operator $T$ is linear.

For each $x\in X$ we have $$ since $({T}_{n}x)$ is . By the principle of uniform boundedness (http://planetmath.org/BanachSteinhausTheorem) we must also have $$.

Then for each $x\in X$ we have

$$\parallel Tx\parallel =\underset{n\to \mathrm{\infty}}{lim}\parallel {T}_{n}x\parallel \le M\parallel x\parallel $$ |

which means that $T$ is . $\mathrm{\square}$

Title | pointwise limit of bounded operators is bounded |
---|---|

Canonical name | PointwiseLimitOfBoundedOperatorsIsBounded |

Date of creation | 2013-03-22 17:32:10 |

Last modified on | 2013-03-22 17:32:10 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 4 |

Author | asteroid (17536) |

Entry type | Corollary |

Classification | msc 46B99 |

Classification | msc 47A05 |