TY - JOUR
AU - Speakman, Jane M. O.
AB - AN ALGEBRAIC CHARACTERISATION OF CONVERGENCE IDEALS JANE M. O. SPEAKMAN 1. Introduction We consider series of non-negative real numbers. The terms of a series will be indexed by Z, the set of positive integers. Even if £#„ diverges there will be a large number of convergent subseries and we define J to be the class {A: A^Z, £/ie/i 0/i < °°}. It is well known that J forms an ideal (the convergence ideal). This means that (i) A e J and B c A => B e</, (ii) A and B e J => AUBE/ and (iii) 0e/. Kakutani [1] has given a necessary and sufficient condition for two series to give rise to the same ideal but the problem of characterising convergence ideals seems to have been open until now. Some necessary conditions have been found by N . G. de Bruijn, P. Erdos, S. Kakutani and R. Rado (unpublished). This paper describes a necessary and sufficient condition for an ideal to be a convergence ideal. The condition is based on the new concept of a. portability-class introduced in §2. Where there is no ambiguity we write S for I a where A is a subset of Z. 
TI - An Algebraic Characterisation of Convergence Ideals
JF - Journal of the London Mathematical Society
DO - 10.1112/jlms/s1-44.1.26
DA - 1969-01-01
UR - https://www.deepdyve.com/lp/wiley/an-algebraic-characterisation-of-convergence-ideals-g0sfOnkMpg
SP - 26
EP - 30
VL - s1-44
IS - 1
DP - DeepDyve
ER -