= 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 <basis>[Hamel basis] $(e_n)$, then
$$
V=\bigcup_{n\geq1}\operatorname{span}(e_1,\ldots,e_n).
$$
Each finite-dimensional subspace is closed and, being proper, nowhere dense. This contradicts the <Baire category theorem>. The polynomials have the countable Hamel basis $1,x,x^2,\ldots$ and are infinite-dimensional, so no norm can make that whole vector space complete.
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