State a version of the Stone-Weierstrass Theorem for real-valued functions on a compact metric space.

Suppose that $K:[0,1]^{2} \rightarrow \mathbb{R}$ is a continuous function. Show that $K(x, y)$ may be uniformly approximated by functions of the form $\sum_{i=1}^{n} f_{i}(x) g_{i}(y)$ with $f_{i}, g_{i}:[0,1] \rightarrow \mathbb{R}$ continuous.

Let $X, Y$ be Banach spaces and suppose that $T: X \rightarrow Y$ is a bounded linear operator. What does it mean to say that $T$ is finite-rank? What does it mean to say that $T$ is compact? Give an example of a bounded linear operator from $C[0,1]$ to itself which is not compact.

Suppose that $\left(T_{n}\right)_{n=1}^{\infty}$ is a sequence of finite-rank operators and that $T_{n} \rightarrow T$ in the operator norm. Briefly explain why the $T_{n}$ are compact. Show that $T$ is compact.

Hence, show that the integral operator $T: C[0,1] \rightarrow C[0,1]$ defined by

$$T f(x)=\int_{0}^{1} f(y) K(x, y) d y$$

is compact.