Definition 2.7 . [00HB]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Definition 2.7.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space. The dual norm of $\lVert\mathord{\cdot}\rVert^{\vee}$ on the dual vector space $V^{\vee}$ is defined by

\forall\ell\in V^{\vee},\ \lVert\ell\rVert^{\vee}:=\sup_{v\in V\setminus\{0\}}\frac{\lvert\ell(v)\rvert}{\lVert v\rVert}.