Definition:Bounded Linear Transformation

Definition
Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a linear transformation.

Then $A$ is said to be a bounded linear transformation iff


 * $\exists c > 0: \forall h \in H: \left\Vert{Ah}\right\Vert_K \le c \left\Vert{h}\right\Vert_H$

In view of Continuity of Linear Transformations, a linear transformation between Hilbert spaces is bounded if and only if it is continuous.

Bounded Linear Operator
If $K$ in fact is equal to $H$, then $A$ is called a bounded linear operator, in line with the definition of linear operator.