# necessary and sufficient conditions for a normed vector space to be a Banach space

Theorem 1 - Let $(X,\parallel \cdot \parallel )$ be a normed vector space^{}. $X$ is a Banach space^{} if and only if every absolutely
convergent series in $X$ is convergent, i.e., whenever $$ ${\sum}_{n}{x}_{n}$ converges in $X$.

Theorem 2 - Let $X,Y$ be normed vector spaces, $X\ne 0$. Let $B(X,Y)$ be the space of bounded operators^{} $X\u27f6Y$. Then
$Y$ is a Banach space if and only if $B(X,Y)$ is a Banach space.

Title | necessary and sufficient conditions for a normed vector space to be a Banach space |
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Canonical name | NecessaryAndSufficientConditionsForANormedVectorSpaceToBeABanachSpace |

Date of creation | 2013-03-22 17:23:04 |

Last modified on | 2013-03-22 17:23:04 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 6 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46B99 |