Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 22H / c

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 $P={\bigcup }_{m\ge 0}{P}_m$. 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. No norm makes the space of all polynomials complete.