Equivalence of Definitions of Norm of Linear Functional/Corollary

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

For all $h \in H$, the following inequality holds:


 * $\left|{Lh}\right| \le \left\|{L}\right\| \left\|{h}\right\|$

Proof
If $h = \mathbf 0$ there is nothing to prove.

Let $h \ne \mathbf 0$.

By the definition of the supremum:
 * $\dfrac{\left|{Lh}\right|} {\left\Vert{h}\right\Vert} \le \left\Vert{L}\right\Vert_3 = \left\Vert{L}\right\Vert$

whence:
 * $\left|{Lh}\right| \le \left\|{L}\right\| \left\|{h}\right\|$