> [!definition] > > Let $E, F$ be [[Locally Convex Topological Vector Space|locally convex topological vector spaces]] and $L(E, F)$ be the space of [[Continuous Linear Map|continuous linear maps]] from $E$ to $F$. For each [[Bounded Set|bounded set]] $B \subset E$ and continuous [[Seminorm|seminorm]] $\rho: F \to \real^+$, the mapping > $ > [\cdot]_{B, \rho}: L(E, F) \to \real^+ \quad T \mapsto \sup_{x \in B}\rho(Tx) > $ > is a seminorm. The topology on $L(E, F)$ defined by seminorms of this form is the **uniform operator topology**. If $E, F$ are [[Normed Vector Space|normed]], then the uniform operator topology is induced by the operator norm.