Suppose the real numbers {a_k} are the terms of a conditionally convergent series. Let C({a_k}) denote the set of all sums of rearrangements of this series. A famous theorem of Riemann shows that, in this particular circumstance, C({a_k}) = R, the set of real numbers.(adsbygoogle = window.adsbygoogle || []).push({});

Various generalizations of this result exist for Banach spaces, but I'm interested in the special case of complex series. In particular, if the complex numbers {a_k} are the terms of a conditionally convergent series, then C({a_k}) is an affine subspace of the complex plane.

I was told this informally by a professor, but the result is completely fascinating to me, so I've been trying to find a proof, either online or by working one out myself. Well, so far I have failed in both endeavors!

I was wondering if anyone had any insight into this problem, either a reference or an idea on how to proceed in proving it.

One thing to note is that given an affine subspace of the plane, one can construct a series with terms {a_k} such that C({a_k}) is that particular subspace. For instance, if {alpha,beta} is a basis of the complex plane and {a_k} and {b_k} are *real* numbers that are the terms of two conditionally convergent series, then C({alpha * a_k + beta * a_k}) is the plane by Riemann's theorem as stated above for series with real terms.

In short, it's evident that any affine subspace can be achieved in the form C({a_k})--the significant part is that C({a_k}) must always have this form! (Of course, this is provided that the associated series is conditionally convergent.)

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# Riemann's theorem for complex series

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