Definition:Norm/Bounded Linear Functional

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

Definition 4
These definitions are equivalent, as proved in Equivalence of Definitions of Norm of Linear Functional.

As a consequence of definition $(4)$, have for all $h \in H$ that $\left\vert{Lh}\right\vert \le \left\Vert{L}\right\Vert \left\Vert{h}\right\Vert$.

As $L$ is bounded, it is assured that $\left\|{L}\right\| < \infty$.

Also see

 * Equivalence of Definitions of Norm of Linear Functional

Special cases

 * Definition:Hilbert Space
 * Definition:Bounded Linear Functional
 * Definition:Norm/Bounded Linear Transformations, of which this is a special case.