Absolute convergence in normed space

Question

1. There is example when convergent sequence does not implies absolute convergence.

2. In wikipedia there is written that for absolute convergence we need norm. So we talk about absolute convergence only in the norm space.

Question. How to link this and the example above where we see that convergence does not implies absolute convergence? My thoughts: if we talk about absolute convergence then it implies that we talk about normed space. In normed space convergence implies absolute convergence. What do I misunderstand?

EDIT: An idea that in normed space convergence implies absolute convergence was taken from this paper.

Answer 1

In order to define partial sums, we need to have some notion of addition. Then, once we have partial sums, we can define the limit of a series in terms of the convergence of the partial sums. If we define absolute convergence in terms of the absolute value of each term, that requires that the absolute value function be defined. Norms are a generalization of the absolute value.

That is, if we have a function $N$, we can ask whether $\lim_{k \rightarrow \infty}\sum_{n=1}^kN(a_n)$ exists. If $N$ is a norm, then convention is that $||x||$ represents $N(x)$, so the preceding expression becomes $\lim_{k \rightarrow \infty}\sum_{n=1}^k||a_n||$. When we're talking about a general space, just having a binary operator defined does not allow us to speak of the absolute value (for instance, what's the absolute value of a permutation?).

In normed space convergence implies absolute convergence.

No, the example of $\frac {(-1)^n}n$ shows that in a normed space, convergence does not imply absolute convergence. That is, in spaces with a norm, "convergence implies absolute convergence" is false. In spaces without a norm, "convergence implies absolute convergence" is meaningless, because the entire concept of "absolute convergence" depends on a norm for its definition.