State the axioms for a norm on a vector space. Show that the usual Euclidean norm on $\mathbb{R}^{n}$,

$$\|x\|=\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)^{1 / 2}$$

satisfies these axioms.

Let $U$ be any bounded convex open subset of $\mathbb{R}^{n}$ that contains 0 and such that if $x \in U$ then $-x \in U$. Show that there is a norm on $\mathbb{R}^{n}$, satisfying the axioms, for which $U$ is the set of points in $\mathbb{R}^{n}$ of norm less than 1 .