Rearrangement and Subseries Numbers

Abstract

By rearranging the terms of a conditionally convergent series we can make it assume a different limit or even diverge. Similarly we could do so by taking a subseries of a conditionally convergent series. The recently studied rearrangement and subseries numbers are the least number of permutations or subsets of indices that are needed to change the behaviour of every conditionally convergent series. The rearrangement and subseries numbers are cardinal characteristics (uncountable cardinalities bound from above by the cardinality of the continuum). In this thesis we will investigate their dual cardinal characteristics, that is, the least number of conditionally convergent series needed such that no single permutation or subset of indices alters the behaviour of all of the series simultaneously. We show that most of the results known about the rearrangement and subseries numbers correspond naturally to dual statements about their dual cardinal characteristics. Additionally we formulate the subseries numbers in an alternative way that gives rise to some new subseries numbers, and prove a few original results about them.