Definition:Bounded Linear Transformation/Bounded Linear Operator

From ProofWiki
Jump to navigation Jump to search

Definition

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.