Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 1
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Theorem
Let $\mathbb K = \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\mathbb K$.
Let $CL$ be the continuous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\struct {\map {CL} {X, \mathbb K}, \norm {\, \cdot \,} }$ is a Banach space.
Proof
- $X' := \map {CL} {X, \mathbb K}$ is the dual space of $X$
So this theorem is the same as Normed Dual Space is Banach Space.
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$\blacksquare$
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?