Definition:Weak Operator Topology
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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.
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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 $\map {CL} {X, Y}$.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X, Y}$. Strong and weak operator topologies on $\map {CL} {X, Y}$