Series in incomplete normed space We have known that "A normed space $X$ is a Banach space if and only if each absolutely convergent series in X converges". We would like to find an **explicitly** incomplete normed space and an **explicitly** series in that space such that the given series is absolutely convergent but not convergent.

This seems to me to be a relatively simple example:

$X=$ set of all real sequences with finite support (i.e., there are only finitely many non-zero elements)

$\|x\|=\sup\limits_{n\in\mathbb N} |x_n|$

Consider the sequence $a_n=(0,\dots,0,\frac1{n^2},0,0,\dots)$ and the series $\sum a_n$ in $X$.

This series is absolutely convergent, since $\sum \|a_n\|= \sum\frac1{n^2}$.

It cannot be convergent in $X$. Take any sequence $x$ with finite support. This means that there is $n_0$ such that $x_n=0$ for each $n\ge n_0$. If $s_n=\sum\limits_{k=1}^n a_k$ denotes the $n$-th partial sum, we have $$\|s_n-x\| \ge \frac1{n_0^2}$$ for each $n\ge n_0$. So w have $\|s_n-x\|\
ot\to0$ and $s_n\
ot\to x$.