= Solution
Suppose for contradiction that the <vector space of univariate polynomials> $P$ were complete in the given <norm>. For $n\geq0$, let
$$
P_n=\{p\in P:\deg p\leq n\}.
$$
Each $P_n$ is finite-dimensional and therefore closed by the fact that a <finite-dimensional subspace is closed>. It is a proper linear subspace of $P$, so it has empty interior: if a linear subspace contained any open ball, translation and scalar multiplication would make it contain every vector. Hence every $P_n$ is a <nowhere dense set>.
But every polynomial has finite degree, so
$$
P=\bigcup_{n=0}^{\infty}P_n.
$$
This writes the supposed complete metric space $P$ as a countable union of nowhere-dense closed subsets, contradicting the <Baire category theorem>. Thus no norm can make $P$ complete. Equivalently, this is the countable-basis obstruction that a <Banach space has uncountable Hamel dimension>.
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