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:


 * $\size {L h} \le \norm L \norm h$

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

Let $h \ne \mathbf 0$.

By the definition of the supremum:
 * $\dfrac {\size {L h} } {\norm h} \le \norm L_3 = \norm L$

whence:
 * $\size {L h} \le \norm L \norm h$