Permutations, Signs, and Sum Ranges

The sum range SR[x; X], for a sequence x = (x(n))(n?N) of elements of a topological vector space X, is defined as the set of all elements s ? X for which there exists a bijection (=permutation) p : N? N, such that the sequence of partial sums (?(n)(k)=1(p(k))( x))(n?)N converges to s. The sum range problem consists of describing the structure of the sum ranges for certain classes of sequences. We present a survey of the results related to the sum range problem in finite-and infinite-dimensional cases. First, we provide the basic terminology. Next, we devote attention to the one-dimensional case, i.e., to the Riemann-Dini theorem. Then, we deal with spaces where the sum ranges are closed affine for all sequences, and we include some counterexamples. Next, we present a complete exposition of all the known results for general spaces, where the sum ranges are closed affine for sequences satisfying some additional conditions. Finally, we formulate two open questions.