Definition:Convergent Sequence in Uniform Operator Topology
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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 $\map {CL} {X, Y}$