(i) State the Stone-Weierstrass theorem for complex-valued functions. Use it to show that the trigonometric polynomials are dense in the space $C(\mathbb{T})$ of continuous, complexvalued functions on the unit circle $\mathbb{T}$ with the uniform norm.

Show further that, for $f \in C(\mathbb{T})$, the $n$th Fourier coefficient

$$\widehat{f}(n)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(e^{i \theta}\right) e^{-i n \theta} d \theta$$

tends to 0 as $|n|$ tends to infinity.

(ii) (a) Let $X$ be a normed space with the property that the series $\sum_{n=1}^{\infty} x_{n}$ converges whenever $\left(x_{n}\right)$ is a sequence in $X$ with $\sum_{n=1}^{\infty}\left\|x_{n}\right\|$ convergent. Show that $X$ is a Banach space.

(b) Let $K$ be a compact metric space and $L$ a closed subset of $K$. Let $R: C(K) \rightarrow$ $C(L)$ be the map sending $f \in C(K)$ to its restriction $R(f)=f \mid L$ to $L$. Show that $R$ is a bounded, linear map and that its image is a subalgebra of $C(L)$ separating the points of

Show further that, for each function $g$ in the image of $R$, there is a function $f \in C(K)$ with $R(f)=g$ and $\|f\|_{\infty}=\|g\|_{\infty}$. Deduce that every continuous, complexvalued function on $L$ can be extended to a continuous function on all of $K$.