Convergence of Series in Banach Spaces

Abstract

It is essential for understanding the concepts of bases of different types to have some knowledge about the various definitions of convergence of a series and to stress its hierarchy and its interconnections. This is done in the first paragraph. There we also present the proof of the important Orlicz- Pettis theorem which claims the equivalence of weak and strong unconditional convergence of series in Banach spaces. The second paragraph contains Riemann’s theorem which asserts that absolute and unconditional convergence of series in finite dimensional vector spaces are the same, and the famous Dvoretzky-Rogers theorem. The latter states the existence in every infinite dimensional Banach space of an unconditional series which is not absolutely convergent, a fact, which has been conjectured for about twenty years and which has been settled down by Dvoretzky and Rogers in 1950.