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?

  • $\begingroup$ Do you have a source saying this can't be done? $\endgroup$ – user137794 Sep 19 '15 at 17:24
  • $\begingroup$ 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 $\endgroup$ – Boby Sep 19 '15 at 17:26

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

  • 1
    $\begingroup$ Thanks. You see, I thought the reason was because absolute converges can not be defined. $\endgroup$ – Boby Sep 19 '15 at 17:37

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