Let $U$ be a vector space. Define what it means for two norms $\|\cdot\|_{1}$ and $\left\|_{\cdot} \cdot\right\|_{2}$ on $U$ to be Lipschitz equivalent. Give an example of a vector space and two norms which are not Lipschitz equivalent.

Show that, if $U$ is finite dimensional, all norms on $U$ are Lipschitz equivalent. Deduce that a finite dimensional subspace of a normed vector space is closed.

Show that a normed vector space $W$ is finite dimensional if and only if $W$ contains a non-empty open set with compact closure.