Rearrangements of Series

Abstract

The present work is devoted to the problems related to the rearrangements of series in metrizable topological vector spaces. The idea goes back to Dirichlet (most probably, he was the first who discovered the phenomenon that the sum of a scalar series may depend on the rearrangement of its terms), Riemann, P. Levy, E. Steinitz, Banach, Kolmogorov, and others.

Two main directions in this area, attracting the interest of many researchers, are considered. The first one concerns the problem of the structure of the sum range of conditionally convergent series. The other is the problem of the existence of an almost sure convergent rearrangement of a functional series, including some classical problems on the convergence of Fourier series.

This book consists mainly of material included in the Ph.D. thesis and some recent works of the author and his colleagues.