# boundedness in a topological vector space generalizes boundedness in a normed space

Boundedness in a topological vector space^{} is a generalization^{} of boundedness in a normed space^{}.

Suppose $(V,\parallel \cdot \parallel )$ is a normed vector space over $\u2102$, and suppose $B$ is bounded^{} in the sense of the parent entry. Then for the unit ball

$$ |

there exists some $\lambda \in \u2102$ such that $B\subseteq \lambda {B}_{1}(0)$. Using this result (http://planetmath.org/ScalingOfTheOpenBallInANormedVectorSpace), it follows that

$$B\subseteq {B}_{|\lambda |}(0).$$ |

Title | boundedness in a topological vector space generalizes boundedness in a normed space |
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Canonical name | BoundednessInATopologicalVectorSpaceGeneralizesBoundednessInANormedSpace |

Date of creation | 2013-03-22 15:33:29 |

Last modified on | 2013-03-22 15:33:29 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Result |

Classification | msc 46-00 |