Definition:Strong Operator Topology

Definition

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

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

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

Let $F = \set {S \stackrel {p_x} \mapsto \norm {Sx - Tx} : \map {CL} {X, Y} \to \R : x \in X : T \in \map {CL} {X, Y}}$ 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 strong operator topology on $\map {CL} {X, Y}$.

Also see

• Results about the strong operator topology can be found here.

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 $\map {CL} {X, Y}$