# Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody's answering this question so I'll try it here. This is really a reference request: Has a certain kind of proof ever been used?

• A series $\displaystyle\sum_n a_n$ converges absolutely if $\displaystyle\sum_n |a_n|<\infty$.
• It converges unconditionally if it converges to a finite number and all of its rearrangements converge to that same number.

For series of real numbers these are equivalent.

There are many proofs of absolute convergence of particular series, and unconditional convergence follows.

My question is whether there are any known direct proofs of unconditional convergence without deducing it from absolute convergence? And are there cases where that method is preferable? Or where absolute convergence was proved by deducing it from unconditional convergence (which would then have to be proved by some other method)? Perhaps where that was the only readily available way to do it?

(Recently I thought I was close to having one of those, and possibly I was, but I decided to do it by proving absolute convergence instead.)

• In a sense the gap between absolute convergence and unconditional convergence is so small it would be difficult to say that one is using one and not the other. – Ryan Budney Sep 1 '15 at 22:25
• Unconditional convergence in Banach spaces is a big research topic. Mathscinet gives hundreds of titles. This strongly suggests that there are some other methods to prove it, besides absolute convergence:-) – Alexandre Eremenko Sep 1 '15 at 23:01
• @RyanBudney : I'm talking about ways of proving something. I had a series whose sum I could easily show is invariant under a very broad class of permutations of the indices without thinking of the sum of the absolute values, and I briefly that it might be a short step from there to showing the same is true of ALL permutations of the indices, still without thinking about the sum of the absolute values. I ultimately decided to go with the old-fashioned way, but it raised this question. ${}\qquad{}$ – Michael Hardy Sep 1 '15 at 23:45
• @AlexandreEremenko : So are those methods of value for some series whose terms are real numbers? ${}\qquad{}$ – Michael Hardy Sep 1 '15 at 23:46
• @Michael Hardy: I don't think so: did not you write yourself that in dimension 1 unconditional convergence is equivalent to absolute convergence? – Alexandre Eremenko Sep 2 '15 at 1:58