The closed unit ball of $C([0,1])$ is not compact

Let \(C([0,1])\) be the space of continuous functions \([0,1]\to\RR\) with the norm \(\norm{f}_\infty\coloneqq\sup_{x\in[0,1]}\abs{f(x)}\). The closed unit ball \(B\coloneqq\theset{f\in C([0,1]) : \norm{f}_\infty\leq1}\) is not compact.