ON THE UNCONDITIONAL CONVERGENCE IN SEMI-ORDERED LINEAR SPACES
Ichirô Amemiya · Hokkaido Mathematical Journal · 1956
In continuous semi-ordered linear spaces, an unconditionally conver-$gent^{I)}$ series is not, in general, absolutely convergent.In this paper we shall give on the one hand, a sufficient condition of the space for this with some examples of the spaces satisfying it; the space of M- type,2) the space of all functions on a set, the sequence space $c_{0}^{3)}$ and etc.. On the other hand, we shall show in every infinite-dimensional space of $L_{p}$ -type4) there exists a series which is unconditionally and not absolutely convergent.On account of it, we shall give a necessary and sufficient condition for monotone complete5) normed spaces under which two notions of convergence are equivalent.Let $R$ be a continuous semi-ordered linear space throughout the paper.