In the classical theory (3) due to Knopp, Agnew and others, the core K(x) of a sequence x = {ξn} of complex numbers is defined by where En(x) is the smallest closed convex set containing all ξk with k ≥ n. A matrix transformation T is said to be a core-preserving transformation if

holds for all sequences x. T is core-preserving for bounded sequences if (1·1) holds for all bounded sequences x. It is readily proved that K(x) is the set of complex numbers ζ such that

for all complex numbers α (). Now is a sub-additive, positive-homogeneous real-valued functional defined on the vector space of bounded complex sequences. This suggests the construction of an abstract theory on the following lines.