Absolute and Unconditional Convergence in Normed Linear Spaces

Abstract

is called absolutely convergent if x | x\l1 a xy to be convergent for every choice of the signs. There are several other equivalent definitions; most of these have been discussed by T. H. Hildebrandt. It is clear that if B is of finite (linear) dimension then (1) is unconditionally convergent if and only if it is absolutely convergent. The problem of finding the spaces for which these two types of convergence are equivalent is mentioned by S. Banach.2 The primary aim of this note is to settle this problem by proving the following result. THEOREM 1. The unconditionally convergent series coincide with the absolutely convergent series if and only if the space B is of finite dimension. Here the only non-trivial assertion is that, if B is of infinite dimension, there is a series (1), which is unconditionally but not absolutely convergent. It is easy to give examples of such series in Hilbert space and similar examples have been given3 for all the usually encountered infinitely dimensional Banach spaces. Interesting partial results on the problem solved by Theorem 1 have been established by M. E. Munroe4 and S. Karlin.5 The two last mentioned papers treat also some related problems and give various consequences of Theorem 1. Our method of proof yields not only Theorem 1 but also the following result.