> [!definition] > > Let $E$ be a [[Locally Convex Topological Vector Space|locally convex topological vector space]], then $E$ is **bornologic** if the following equivalent conditions hold: > > 1. Let $M \subset E$ be a [[Convexity|convex]], [[Balanced Set|balanced]] set. If $M$ absorbs every [[Bounded Set|bounded set]] in $E$, then $M$ is a [[Neighbourhood|neighbourhood]] of $0$. > 2. Every [[Seminorm|seminorm]] on $E$ that is bounded on bounded sets is [[Continuity|continuous]]. > > *Proof*. Let $[\cdot]: E \to \real^+$ be a seminorm that is bounded on bounded sets. Let $M = \bracs{x \in E: [x] < 1}$, then $M$ is convex and balanced. Since $[\cdot]$ is bounded on bounded sets, for every $B \subset E$ bounded, there exists $R \ge 0$ such that $B \subset RM$, so $M$ absorbs all bounded sets in $E$. > > If $(1)$ holds, then $M \in \cn^o(0)$, and $[\cdot]$ is continuous. > > If $(2)$ holds, then taking the [[Gauge|Minkowski functional]] of any $M \subset E$ satisfying $(1)$ yields the desired result.