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 $B\subseteq \lambda B_1(0)$. Usingthis result (http://planetmath.org/ScalingOfTheOpenBallInANormedVectorSpace), it follows that

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