# scaling of the open ball in a normed vector space

Let $V$ be a vector space over a field $F$ (real or complex), and let $\parallel \cdot \parallel $ be a norm on $V$. Further, for $r>0$, $v\in V$, let

$$ |

Then for any non-zero $\lambda \in F$, we have

$$\lambda {B}_{r}(v)={B}_{|\lambda |r}(\lambda v).$$ |

The claim is clear for $\lambda =0$, so we can assume that $\lambda \ne 0$. Then

$\lambda {B}_{r}(v)$ | $=$ | $$ | ||

$=$ | $$ | |||

$=$ | $$ | |||

$=$ | ${B}_{|\lambda |r}(\lambda v).$ |

Title | scaling of the open ball in a normed vector space^{} |
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Canonical name | ScalingOfTheOpenBallInANormedVectorSpace |

Date of creation | 2013-03-22 15:33:25 |

Last modified on | 2013-03-22 15:33:25 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 7 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 46B99 |