Let $K$ be a compact Hausdorff space, and let $C(K)$ denote the Banach space of continuous, complex-valued functions on $K$, with the supremum norm. Define what it means for a set $S \subset C(K)$ to be totally bounded, uniformly bounded, and equicontinuous.

Show that $S$ is totally bounded if and only if it is both uniformly bounded and equicontinuous.

Give, with justification, an example of a Banach space $X$ and a subset $S \subset X$ such that $S$ is bounded but not totally bounded.