Definition:Convergent Sequence 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 $\sequence {T_n}_{n \mathop \in \N}$ is a sequence in $\map {CL} {X, Y}$.

Suppose $T \in \map {CL} {X, Y}$.

Then $\sequence {T_n}_{n \mathop \in \N}$ is said to converge to $T$ in $\tau$ if:

$\ds \lim_{n \mathop \to \infty} \norm {T_n - T} = 0$

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