Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 2
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Theorem
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space.
Let $CL$ be the continuous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\struct {\map {CL} {X, X}, \norm{\, \cdot \,}}$ is a Banach space.
Proof
Take $Y = X$ in Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X, Y}$. When is $\map {CL} {X, Y}$ complete?