# 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 $(V,\|\cdot\|)$ is a normed vector space over $\mathbbmss{C}$, and suppose $B$ is bounded in the sense of the parent entry. Then for the unit ball

 $B_{1}(0)=\{v\in V:\|v\|<1\}$

there exists some $\lambda\in\mathbbmss{C}$ 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 BoundednessInATopologicalVectorSpaceGeneralizesBoundednessInANormedSpace 2013-03-22 15:33:29 2013-03-22 15:33:29 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Result msc 46-00