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.



$\blacksquare$

Sources