Euclidean norm duality (${\sup }_{\Vert v{\Vert }_2\le 1}|x^{T}v|=\Vert x{\Vert }_2$)

The Euclidean norm is self-dual:The upper bound is Cauchy-Schwarz inequality, and equality holds at $v=x/\Vert x{\Vert }_2$ when $x\ne 0$.