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series

Type of Math Object: 
Definition
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This entry is too dry for me. Anybody willing to write something about the subject above? Cheers

For real numbers it only makes sense to consider countable index sets. If you take an uncountable set of positive real numbers and you want to define their "sum", it turns out that it is always infinite.. and the argument is that you can always find a countable subset whoose sum diverges (in the usual sense)..

There are many ways to see this, and one of them is:

Suppose you have an uncountable set of positive numbers X.

Consider the following subsets of R^+ (the set of positive real numbers): [1, \infty[; [1/2,1]; [1/3, 1]; [1/4, 1] ... [1/n, 1] ...

These subsets cover R^+, so at least one of them ([1/n, 1] for example) has an infinite number of elements of X. Now take a countable set of this infinite set and consider its series. All its elements are greater than 1/n, so the series diverges in the usual sense.

Yeah, an entry about this would be nice.. If I have time and nobody else does it I'll write it later today.

I may write one in the next couple days. I would define it as follows (and I'm basically just ripping off Rudin): if A is any set and f:A-[0,infty] is a function into the non-negative extended real numbers, then the "sum"

\sum_{a\in A}f(a)

is defined to be

\sup\{sum_{a\in F}f(a):F\subseteq A,F finite\}. If you do it this way, then it turns out that the above "sum" is equal to the integral of f over A with respect to the counting measure. Then you can (naturally?) generalize to complex-valued functions in l^1(A) by defining the unordered sum to be the Lebesgue integral of the function over A. I suppose this may be a bit contrived, but I kind of like it.

> For real numbers it only makes sense to consider
> countable index sets.

Except when all but a countable number of them are zero,
in which case we can define their sum to be the sum of
the countably many non-zero elements. This doesn't
contradict anything said above, just a way of generalizing
the definition.

The reason any of this is interesting is because we encounter
situations where such a generalized definition allowing
sums over uncountable sets is useful in certain situations.
For instance, in non-separable Hilbert (or, more generally,
Banach) spaces, we can have an uncountable number of basis
vectors and can express any vector in the space as a sum
over these basis elements if we define sums over uncountable
sets in the way described above.

> Yeah, an entry about this would be nice.. If I have time and
> nobody else does it I'll write it later today.

I already wrote an entry on this topic two years ago:

http://planetmath.org/?op=getobj;from=objects;id=7698

Yeah, that's it! :) Now it would be nice to also have the definition Keenan was referring to..

The definition of the sum as the supremum of sums over finite
subsets is to be found in the entry I wrote. The formulation as
Lesbegue integrals with a discrete measure is in the entry Boris
wrote:

http://planetmath.org/encyclopedia/SupportOfIntegrableFunctionWithRespec...

Thanks for pointing those entries out.

Thank you all for your replies. I think it'd be nice to see a link to the old (and beautiful) entry by rspuzio directly on the page about series, imho. Or maybe even merge it into it?

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