Dual norm

Let \(X\neq\theset{0}\) be a normed vector space over \(\CC\), and let \(\dualof{X}\) be the space of continuous linear functionals \(X\to\CC\). For \(L \in \dualof{X}\), the dual norm or operator norm of \(L\) is \[ \norm{L}_{\dualof{X}} \coloneqq \sup_{ \substack{x\in X \\ \norm{x} = 1} } \abs{L(x)} = \sup_{ \substack{x\in X \\ \norm{x} \leq 1} } \abs{L(x)}. \]