Space of Compact Linear Transformations is Banach Space

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Theorem

Let $H, K$ be Hilbert spaces, and let $B_0 \left({H, K}\right)$ be the space of compact linear transformations from $H$ to $K$.

Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$.


Now $B_0 \left({H, K}\right) \subseteq K^H$, the set of mappings from $H$ to $K$.

Therefore, $B_0 \left({H, K}\right)$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$).

Let $\left\Vert{\cdot}\right\Vert$ denote the norm on bounded linear operators.


Then $\left\Vert{\cdot}\right\Vert$ is a norm on $B_0 \left({H, K}\right)$.

Furthermore, $B_0 \left({H, K}\right)$ is a Banach space with respect to this norm.


Proof


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