Compact Linear Transformation is Bounded

Theorem
Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be Hilbert spaces.

Let $\map {B_0} {\HH, \KK}$ be the space of compact linear transformations $\HH \to \KK$.

Let $\map B {\HH, \KK}$ be the space of bounded linear transformations $\HH \to \KK$.

Let $T \in \map {B_0} {\HH, \KK}$.

Then $T \in \map B {\HH, \KK}$.

That is:
 * $\map {B_0} {\HH, \KK} \subseteq \map B {\HH, \KK}$