Definition:Bounded Linear Transformation/Bounded Linear Operator

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Let $\HH$ be a Hilbert space.

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

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

$\exists c > 0: \forall h \in \HH: \norm {A h}_\HH \le c \norm h_\HH$

where $\norm {\,\cdot\,}_\HH$ denotes the norm of $A$.

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