On a summability theorem of Berg, Crawford and Whitley

Abstract

We recall that a matrix A is said to sum a sequence x if Axε c, the space of all convergent sequences, and that A is conservative if it sums every convergent sequence. If A is conservative, A defines a continuous linear operator on c. Berg (2), Crawford (3)and Whitley (9) have proved the following theorem:Theorem 1. A conservative matrix sums no bounded divergent sequence if and only if,considered as an operator on c, it is range closed and has finite-dimensional null-spac