Permutations, Signs, and Sum Ranges

[Abstract:] The sum range SR[x; X], for a sequence x = (xn)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) π : N → N, such that the sequence of partial sums (∑nk=1xπ(k))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.The second author was supported by the Spanish Agencia Estatal de Investigación (AEI), grant MTM2016-79422-P, cofinanced by the European Regional Development Fund (EU)