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 definition To 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:
- $\dfrac {\size {L v} } {\norm v} \le \norm L_3 = \norm L$
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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 $3.3$