Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 22H / a / Solution

If a subspace $W\subset V$ contains a ball about some $w\in W$, translation gives a ball about zero in $W$, and scaling then gives every vector of $V$. Thus every proper subspace has empty interior.

If a complete infinite-dimensional $V$ had a countable Hamel basis $(e_n)$, thenEach finite-dimensional subspace is closed and, being proper, nowhere dense. This contradicts the Baire category theorem. The polynomials have the countable Hamel basis $1,x,x^2,\dots$ and are infinite-dimensional, so no norm can make that whole vector space complete.