(i) Prove Riesz's Lemma, that if $V$ is a normed space and $A$ is a vector subspace of $V$ such that for some $0 \leqslant k<1$ we have $d(x, A) \leqslant k$ for all $x \in V$ with $\|x\|=1$, then $A$ is dense in $V$. [Here $d(x, A)$ denotes the distance from $x$ to $A$.]

Deduce that any normed space whose unit ball is compact is finite-dimensional. [You may assume that every finite-dimensional normed space is complete.]

Give an example of a sequence $f_{1}, f_{2}, \ldots$ in an infinite-dimensional normed space such that $\left\|f_{n}\right\| \leqslant 1$ for all $n$, but $f_{1}, f_{2}, \ldots$ has no convergent subsequence.

(ii) Let $V$ be a vector space, and let $\|\cdot\|_{1}$ and $\|.\|_{2}$ be two norms on $V$. What does it mean to say that $\|\cdot\|_{1}$ and $\|.\|_{2}$ are equivalent?

Show that on a finite-dimensional vector space all norms are equivalent. Deduce that every finite-dimensional normed space is complete.

Exhibit two norms on the vector space $l^{1}$ that are not equivalent.

In addition, exhibit two norms on the vector space $l^{\infty}$ that are not equivalent.