Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 22H (Linear Analysis)

The Baire category theorem states that a complete metric space is a Baire space: the intersection of countably many open dense sets is dense. Equivalently, a nonempty open subset cannot be a countable union of nowhere dense sets.

Let $U_n$ be open dense and $O$ any nonempty open set. Choose a closed ball $B_1$ of positive radius at most $1/2$ inside $O\cap U_1$. Inductively choose $B_{n+1}$ inside the interior of $B_n$ intersected with $U_{n+1}$, with positive radius at most $2^{-(n+1)}$. Density makes this intersection nonempty, and openness supplies a sufficiently small closed ball. The centres form a Cauchy sequence because all later centres lie in $B_n$, whose diameter tends to zero. Completeness gives a limit $x$. Since each ball is closed and contains all later centres, $x\in B_n$ for every $n$. Thus $x\in O\cap {\bigcap }_n{U}_n$, proving density. Applying this to complements of the closures of nowhere dense sets gives the equivalent formulation.

If a linear subspace $V$ of a normed space contains an open ball $B(v,r)$, then $v\in V$ and subtraction gives $B(0,r)\subseteq V$. For any $x\in X$, choose a positive scalar $t$ with $\Vert tx\Vert <r$. Then $tx\in V$ and hence $x\in V$. Thus $V=X$. Every proper linear subspace has empty interior.

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.