Definition:Bounded Linear Functional

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Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$.

Then $L$ is said to be a bounded linear functional iff

$\exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$

In view of Continuity of Linear Functionals, a linear functional on a Hilbert space is bounded if and only if it is continuous.

Also see