Past exam of the analysis course of the University of Oxford / 2011 / 1 / b / iii / Solution

Write $q(x)={\sup }_k|{\sum }_{n=1}^k{x}_n|$. This is indeed a norm: the triangle inequality and homogeneity follow from the scalar absolute value, and vanishing of every partial sum forces every coordinate to vanish. It is finite on ${\ell }^1$ because $q(x)\le \Vert x{\Vert }_1$.

Let $a_n=(-1{)}^{n+1}/n$ and let $a^{(N)}$ be its first $N$ entries followed by zeros. The alternating-series bound gives, for $M>N$,Thus these truncations are Cauchy for $q$. But $|x_n|\le 2q(x)$ for every coordinate, since $x_n$ is the difference of two consecutive partial sums. Any $q$-limit would consequently have coordinates $a_n$. This sequence is not in ${\ell }^1$. Hence