Let $K$ be a compact Hausdorff space.

(a) State the Arzelà-Ascoli theorem, and state both the real and complex versions of the Stone-Weierstraß theorem. Give an example of a compact space $K$ and a bounded set of functions in $C(K)$ that is not relatively compact.

(b) Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be continuous. Show that there exists a sequence of polynomials $\left(p_{i}\right)$ in $n$ variables such that

$$B \subset \mathbb{R}^{n} \text { compact }\left.\left.\Rightarrow \quad p_{i}\right|_{B} \rightarrow f\right|_{B} \text { uniformly }$$

Characterize the set of continuous functions $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ for which there exists a sequence of polynomials $\left(p_{i}\right)$ such that $p_{i} \rightarrow f$ uniformly on $\mathbb{R}^{n}$.

(c) Prove that if $C(K)$ is equicontinuous then $K$ is finite. Does this implication remain true if we drop the requirement that $K$ be compact? Justify your answer.