= Solution
Let $P_m$ be the <linear subspace> of <polynomials> of degree at most $m$. It has finite dimension $m+1$, is proper in $P$, and $P=\bigcup_{m\geq0}P_m$. For any norm on $P$, the restricted norm on $P_m$ is equivalent to its coefficient Euclidean norm: on the coefficient unit sphere the continuous norm has a positive minimum and a finite maximum. Thus $P_m$ is complete. A complete subspace of a <metric space> is closed, since any sequence converging to an ambient limit is Cauchy and has the same limit in the subspace.
Part (b) makes each $P_m$ a closed set with empty interior, hence nowhere dense. If $P$ were a <Banach space>, this countable union would contradict the <Baire category theorem>. \b[No norm makes the space of all <polynomials> complete.]
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