Permutations, Signs, and Sum Ranges

Abstract

[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.