Let $(K, d_K)$ be a compact metric space, and let $\mathcal{F} \subset C(K)$. Then $\mathcal{F}$ is totally bounded in $(C(K), \|\cdot\|_\infty)$ if and only if:

1. $\mathcal{F}$ is **uniformly bounded**: there exists $M > 0$ with $\|f\|_\infty \le M$ for all $f \in \mathcal{F}$. 2. $\mathcal{F}$ is **equicontinuous**: for every $\varepsilon > 0$, there exists $\delta > 0$ such that $d_K(x, y) < \delta$ implies $|f(x) - f(y)| < \varepsilon$ for all $f \in \mathcal{F}$ and all $x, y \in K$.