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 $ℂ$, and suppose $B$ is bounded in the sense of the parent entry. Then for the unit ball

there exists some $\lambda \in ℂ$ 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
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