Equivalence of methods for the summation of double sequences

In 1900 Pringsheim presented the following notion of convergence for double sequences: a double sequence [x][x] converges to LL provided that given ϵ>0ϵ>0 there exists K∈N such that |xk,l−L|<ϵ|xk,l−L|<ϵ whenever k,l>Kk,l>K. Using this definition Robison and Hamilton, in 1926 and 1936, respectively, presented the following definition for the regularity of four dimensional matrices. A four dimensional matrix AA is regular if it maps every bounded convergent sequence into a convergent sequence with the same limit. These notions shall be used to present simple conditions to ensure that Tm,n=∑k,l=0,0m,nam,n,k,lxk,l and σm,n=∑k,l=0,0∞,∞am,n,k,lxk,l are included by convergence.