The supremum of $\abs{L(x)}$ on the unit sphere equips $\dualof{H}$ with a norm.Suggest a slogan
Let \(H\neq\theset{0}\) be a Hilbert space over \(\CC\), and let \(\dualof{H}\) be its dual. Then the operator norm \(\norm{L}_{\mathrm{op}}\coloneqq\sup_{\norm{x}=1}\abs{L(x)}\) is a norm on the vector space \(\dualof{H}\).