Definition:Weak Operator Topology

Definition
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.

Let $\mathbb K = \set {\R, \C}$.

Let $\map {CL} {X, Y}$ be a continuous linear transformation space.

Let $F = \set {S \stackrel {p_{x,\phi} } \mapsto \size {\phi \paren{Sx - Tx} } : \map {CL} {X, Y} \to \R : x \in X : \phi \in \map {CL} {Y, \mathbb K} : T \in \map {CL} {X, Y}}$ be a set of maps.

Let $\tau$ be the weakest topology on $\map {CL} {X, Y}$ such that every $y \in F$ is continuous.

Then $\tau$ is called the weak operator topology on $\map {CL} {X, Y}$.