Definition:Bounded Subset of Topological Vector Space

Definition

Let $\mathbb F \in \set {\R, \C}$.

Let $\struct {V, \tau}$ be a topological vector space over $\mathbb F$.

A subset $B \subseteq V$ is bounded if and only if:

for each $U \in \tau$ such that $\mathbf 0_V \in U$ there is an $\epsilon \in \R_{>0}$ such that:

$\epsilon B \subseteq U$

where:

$\bf 0_V$ denotes the zero vector of $V$

$\epsilon B$ denotes the dilation of $B$ by $\epsilon$