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I was particularly interested in the following:

When I read this proof, everything seemed fine and logical except one detail (the proof is located here).

Right after we prove, that the series $\sum_{k=1}^\infty x_{N_m} - x_{N_m+1}$ converges, there is a statement which tells us that the limit of that series (let's name it $s$) definitely belongs to the initial space $\mathbb{X}$: $s \in \mathbb{X}$.

Why is that? That could probably be very obvious, but unfortunately I can't get it.

Why should a $lim \hspace{2 mm} \sum_{k=1}^\infty (x_{i} - x_{j})$, where $x_i \in \mathbb{X}$ belong to $\mathbb{X}$ itself?

What am I missing? :)

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up vote 5 down vote accepted

Because that is what it means for a series to converge in $X$. It means that the sequence of partial sums converges to an element of $X$. In general, convergence of a sequence means that there is some element of your space to which the sequence converges.

To elaborate a bit, to say that a sequence $(y_n)$ in $X$ converges without adding any qualification is another way to say that the sequence converges in $X$, which means that it has a limit in $X$. That is, there is an element $L$ of $X$ such that for all $\varepsilon\gt0$ there exists an $N$ such that $n\gt N$ implies $\|y_n-L\|\lt\varepsilon$. That is precisely the notion used in the hypothesis at your link, and it is the convergence referred to when we say that Cauchy sequences converge in complete spaces.

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I always thought that convergense of a $series$ means that the partial sum sequence has a finite limit. But the definition of the convergence doesn't state that that limit should belong to the same space as the elements of the series. Or that should be obvious that it SHOULD belong to it? – Yippie-Kai-Yay Dec 20 '10 at 19:55
It really is the definition. In the context of series of real numbers, you get the impression that "finite vs. infinite" is the only problem, and sometimes the fact that convergence of a sequence of real numbers really means convergence to a particular real number isn't in the forefront. But that is what it means. Now trade in the real numbers for a more general space, like a normed vector space, and convergence is defined to mean the same thing; the existence of a limit in the space. – Jonas Meyer Dec 20 '10 at 20:00
That's a nice explanation, thank you. – Yippie-Kai-Yay Dec 20 '10 at 20:01
When we say that a real sequence converges to plus or minus infinity as opposed to not having a limit we are not really working in R, but in its two-point compactification in which plus and minus infinity both exist as actual points. Limits are always defined relative to the ambient space, but by changing the ambient space we change our notion of limit. – Qiaochu Yuan Dec 20 '10 at 20:15

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