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uncountable sums of positive numbers

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Can't this be extended to series of real numbers (maybe negative) as long as the series of |a_i| is convergent (in this sense)?

I think it can but maybe I am wrong.

My guess is that as long as the series is "absolutely convergent" (here in an extended sense), the sup of the series \sum a_i will also exist.

"I think it can but maybe I am wrong."

Well, I am wrong!
the sup \sum a_i will exists but will not be the value of the sum

But it can still be defined.

sum only the positive elements (their sum exists because \sum |a_i| exists

sum only the |negatives| (also exists for the same reason)

subtract the second from the first. Thats the value of \sum a_i.

You could do it for a general Banach space. If (a_i)_{i\in I} is a family of elements from a Banach space X, then for every finite subset F\subset I, define a_F=\sum_{i\in F}a_i. The set of all finite subsets (call it S) of I forms a directed set under inclusion. and hence (a_F)_{F\in S} forms a net in X. So we can say
\sum_{i\in I}a_i=\lim_{F\in S}a_F
if the limit of this net exists.

Something to note is that for a countable set I, this coincides with the usual notion of a summability only if the series converges absolutely.

Here is an equivalent definiton (to bob1s) that uses less terminology.

Sum(c_i,i,I) = S

means

For all eps>0, there exists a finite subset t of I such that for all finite sets T such that t is a subset of T is a subset of I,
|S-(Sum(c_i,i,T))| < eps.

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