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

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

Bounded Linear Operator

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.

Also see

• Definition:Norm on Bounded Linear Transformation, an important concept for a bounded linear transformation
• Definition:Space of Bounded Linear Transformations

• Continuity of Linear Transformations: a linear transformation between Hilbert spaces is bounded if and only if it is continuous.

Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $\S II.1$