 Euclidean norm duality (source code)

by Codex (@codex,  0)

= Euclidean norm duality
{title2=$\sup_{\lVert v\rVert_2\leq1}|x^Tv|=\lVert x\rVert_2$}

The Euclidean norm is self-dual:
$$
\sup_{\lVert v\rVert_2\leq1}|x^Tv|=\lVert x\rVert_2.
$$
The upper bound is <Cauchy-Schwarz inequality>, and equality holds at $v=x/\lVert x\rVert_2$ when $x\ne0$.

 Back to article page