Monotone completeness of normed semi-ordered linear spaces
Sadayuki Yamamuro · Pacific Journal of Mathematics · 1957
Introduction* Let R be a continuous semi-ordered linear space, namely, a semi-ordered linear space where, for any sequence x y ^0 (y=l,2, •)> f\x v exists. 1 R is said to be a normed semi-ordered linear space, if a norm ||α?||(a?e J?) is defined and satisfies the condition: \x\^\y\ implies IMI^IMI in addition to the usual conditions.A norm |a?||(a;e22) on a normed semi-ordered linear space is said to be monotone complete, if, when 0^# v ΐ and sup||# v || < + co , there existsand semi-continuous, if 0^α? v a? implies sup |a? v |=fa?||.It is clear that IV=l VS1 continuity implies semi-continuity.Kantorovitch [4] has proved that, if a norm on 22 is monotone complete and continuous, then it is complete, namely, 22 is a Banach lattice.Nakano [5; Theorem 31.7] has proved that, if a norm on 22 is monotone complete and semi-continuous, then the norm is complete, and, recently, Amemiya [1] has proved that, if a norm on 22 is monotone complete, it is complete.2 In this connection, see also [2].In this paper, we will consider several problems concerning monotone completeness and completeness of normed semi-ordered linear spaces and Nakano spaces.lMonotone completeness of normed semi-ordered linear spaces* In this section, we will consider two problems.As usual, let (c 0 ) be the set of all null-sequences of real numbers.This is a normed semi-ordered linear space by the usual ordering and