space of continuous functions

Once you measure functions with the sup norm, the collection of all continuous functions on a compact set stops being a vague 'bunch of functions' and becomes a single geometric object: a space whose points are functions and whose distance is the worst-case gap. You can then talk about convergence, completeness, and balls of functions exactly as you would for points in R^n.

Let K be a compact metric space (for instance a closed interval [a, b]). The space C(K) consists of all continuous real- (or complex-) valued functions on K, equipped with the sup norm ||f|| = max over K of |f(x)|. This makes C(K) a normed vector space, and convergence in this norm is precisely uniform convergence.

The central theorem is that C(K) is COMPLETE: every uniformly Cauchy sequence of continuous functions converges, in sup norm, to a continuous function. (Completeness comes from the uniform Cauchy criterion, and the limit stays continuous by the uniform limit theorem.) A complete normed space is called a Banach space, so C(K) is a Banach space — the natural home for approximation theorems like Weierstrass's and Stone–Weierstrass.