# Why in normed vector spaces we can define infinite series but in metric space we can not?

We usually define infinite series by partial sums and an inifinite series is said to converge if its partial sum converges.

So, if $X$ is a normed vector spaces and $s_n=x_1+...+x_m$ is a partial sum then infinite series converges to s if \begin{align} \lim_{n \to \infty}||s-s_n||=0. \end{align}

My question why can't the same be done in vector metric spaces? We can still define converges as \begin{align} \lim_{n \to \infty}d(s,s_n)=0 \end{align}

Certainly when a metric is induced by a norm this is not a problem. But why is it a problem in the case when metric can not be defined by a norm?

• Do you have a source saying this can't be done? – user137794 Sep 19 '15 at 17:24
• For example, in Kreyszig the author waits and introduces convergens of infinite series until he defines normed spaces. My guess is because one can not define d absolute converegence – Boby Sep 19 '15 at 17:26

## 1 Answer

You can do this. In fact, people even consider topological vector spaces.

• Thanks. You see, I thought the reason was because absolute converges can not be defined. – Boby Sep 19 '15 at 17:37