Equivalence of Definitions of Norm of Linear Functional/Corollary

Theorem
Let $V$ be a normed vector space, and let $L$ be a bounded linear functional on $V$.

For all $v \in V$, the following inequality holds:


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

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

Let $v \ne \mathbf 0$.

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

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