Definition:Bounded Linear Transformation

Definition

Let $H, K$ be Hilbert spaces.

Let $A: H \to K$ be a linear transformation.

Then $A$ is a bounded linear transformation if and only if

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

Let $H$ be a Hilbert space.

Let $A: H \to H$ be a linear operator.

$A$ is a bounded linear operator if and only if

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

That is, a bounded linear operator is a bounded linear transformation from a Hilbert space to itself.