Space where normal convergence does not imply uniform convergence

Question

We know that if we're in a Banach space, normal convergence implies uniform convergence. Is there an easy counterexample outside Banach spaces ?

Answer 1

If we restrict our attention to constant functions, then "normal convergence implies uniform convergence" just becomes the statement that for any sequence $x_n$ in the normed space $X$, if $\sum \|x_n\| < \infty$ then $\sum x_n$ converges in norm. And this is in fact equivalent to the completeness of $X$, so it is false in every incomplete normed space.

The fact that it is true in complete spaces is well known. Conversely, suppose $X$ is a normed space in which the statement holds. Let $x_n$ be a Cauchy sequence in $X$. Then for every $k$, there exists some $N_k$ such that for all $m,n \ge N_k$, we have $\|x_m - x_n\| \le 1/2^k$. Consider the sequence $y_k = x_{N_{k+1}} - x_{N_k}$, so that $\|y_k\| \le 1/2^k$; then $\sum y_k$ converges. But the $j$th partial sum $\sum_{k=1}^j y_k$ of this series is just $x_{N_j} - x_{N_1}$, so we have shown that the sequence $x_{N_j}$ converges; that is, $x_n$ has a convergent subsequence. Since $x_n$ was already Cauchy, it follows by a standard triangle inequality argument that in fact $x_n$ itself converges.