Absolute Convexity in Spaces of Strongly Summable Sequences

Abstract

The space wp of strongly Cesàro summable sequences of index p > 0 has been investigated by several authors. In [2], Kuttner proved that no Toeplitz matrix could sum all sequences in wP, a result which was extended to coregular matrices by Maddox [5]. In [1], Borwein considered the continuous dual space of wp. The more general space w(p) has also been considered [3, 4], where p = (pk) is a strictly positive sequence. The r-convexity of the spaces w∞(p) and w0(p) was dealt with in a partial way in [8]. In the present note we establish criteria for the rconvexity of some general classes of [A, p]0 and [A, p]∞ spaces (see [6] and [7] for definitions), and in particular we give the necessary and sufficient conditions for the r-convexity of w∞(p) and w0(p). For most of the relevant definitions and notation we refer to [8].