Compact Linear Transformation is Bounded

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Theorem

Let $H, K$ be Hilbert spaces.

Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation.


Then $T$ is also a bounded linear transformation.

That is, $B_0 \left({H, K}\right) \subseteq B \left({H, K}\right)$.


Proof


Sources