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.

See also

• Norm (Linear Transformation), an important concept for a bounded linear transformation
• Space of Bounded Linear Transformations

Sources

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