(i) Define the dual of a normed vector space $(E,\|\cdot\|)$. Show that the dual is always a complete normed space.

Prove that the vector space $\ell_{1}$, consisting of those real sequences $\left(x_{n}\right)_{n=1}^{\infty}$ for which the norm

$$\left\|\left(x_{n}\right)\right\|_{1}=\sum_{n=1}^{\infty}\left|x_{n}\right|$$

is finite, has the vector space $\ell_{\infty}$ of all bounded sequences as its dual.

(ii) State the Stone-Weierstrass approximation theorem.

Let $K$ be a compact subset of $\mathbb{R}^{n}$. Show that every $f \in C_{\mathbb{R}}(K)$ can be uniformly approximated by a sequence of polynomials in $n$ variables.

Let $f$ be a continuous function on $[0,1] \times[0,1]$. Deduce that

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