Finite dimentional topological vector spaces

Question

I was reading the functional analysis book of R.E. Edwards there is a theorem states that " A topological vector spaces is finite dimentional if and only if there exists a precompact neighbourhood of zero " . the " only if " part is ok but how can we tackle the converse part . Anyone any hints ?

Answer 1

Hint: Prove the contrapositive: if $V$ is infinite dimensional then the unit ball is not compact in $V$.

To prove the hint you need to prove the following lemma due to Riesz:

If $V$ is infinite dimensional, then for every proper subspace of $V$, say $X$, there exists $x \not\in X$ such that $\|x\| = 1$ and $\|x-y\|> \frac{1}{2}$ for all $y \in X$.