# quotient norm

Let $V$ be a normed vector space^{} with norm $\parallel \cdot \parallel $. Let $M$ be a closed subspace of $V$ and $V/M$ the
quotient vector space^{}.

The norm $\parallel \cdot \parallel $ induces a norm $\parallel \cdot {\parallel}_{V/M}$ in $V/M$, called the quotient norm, given by

$${\parallel v+M\parallel}_{V/M}:=\underset{u\in v+M}{inf}\parallel u\parallel =\underset{m\in M}{inf}\parallel v+m\parallel $$ |

Theorem - $\parallel \cdot {\parallel}_{V/M}$ is a norm in $V/M$ iff $M$ is closed in $V$.

Title | quotient norm |
---|---|

Canonical name | QuotientNorm |

Date of creation | 2013-03-22 17:22:58 |

Last modified on | 2013-03-22 17:22:58 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Definition |

Classification | msc 46B99 |