Definition:Open Subset in Uniform 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 $\tau$ be the uniform operator topology induced by $\struct {\map {CL} {X, Y}, \norm {\, \cdot \,}}$.

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

$\forall T \in U : \exists \epsilon \in \R_{>0} : \set {S \in \map {CL} {X, Y} : \norm {S - T} < \epsilon} \subseteq U$

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