# Equivalence of Definitions of Norm of Linear Functional/Corollary

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It has been suggested that this page be renamed.In particular: This can't really be a corollary of an equivalence definitionTo discuss this page in more detail, feel free to use the talk page. |

## 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:

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

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## Proof

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

Let $v \ne \mathbf 0$.

By the definition of the supremum:

- $\dfrac {\size {L v} } {\norm v} \le \norm L_3 = \norm L$

This article, or a section of it, needs explaining.In particular: What is this $\norm L_3$You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

whence:

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

$\blacksquare$

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem: Proposition $3.3$