Space of continuous functions - Imported Knowlpedia Content

The space of continuous functions on a topological space $X$ is

C(X)=\{f:X\to\mathbb{R} \mid f \text{ is continuous}\},

where continuity is in the sense of a continuous map. It is naturally a vector space under pointwise addition and scalar multiplication.

Remarks

On domains where continuous functions are bounded (for example, on a compact interval), $C(X)$ can be equipped with the supremum norm and the associated uniform metric, linking function-space topology to uniform convergence. Theorems such as Arzelà–Ascoli and Stone–Weierstrass are statements about subsets of $C(X)$ .

Examples