Definition:Open Subset in 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 $\tau$ be the weak operator topology on $\map {CL} {X, Y}$.

Suppose $U \subseteq \map {CL} {X, Y}$ is an open subset such that:

$\forall T \in U : \exists \epsilon \in \R_{>0} : \exists n \in \N : \forall k \in \N : k \le n : \exists x_k \in X : \exists \phi_k \in \map {CL} {Y, \mathbb K} : \set {S \in \map {CL} {X, Y} : \forall i \in \N : i \le n : \size {\phi_i \paren{S x_i - T x_i} } < \epsilon } \subseteq U$

Then $U$ is called an open subset in $\tau$.

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