Equivalence of Definitions of Norm of Linear Functional/Corollary

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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:

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

$\blacksquare$


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