The dual space $\dualof{X}$ with the operator norm is a Banach space

Let \(X\) be a normed vector space over \(\mathbb K\in\theset{\RR,\CC}\), and let \(\dualof{X}\) be the vector space of continuous linear functionals \(X\to\mathbb K\) with the operator norm \(\norm{L}_{\text{op}}\coloneqq\sup_{x\in X,\ \norm{x}\leq1}\abs{L(x)}\). Then \((\dualof{X},\norm{\cdot}_{\text{op}})\) is a Banach space.