PRODUCT SPACES OF SEMI-ORDERED LINEAR SPACES
Hidegorô Nakano · Hokkaido Mathematical Journal · 1953
we obtain a product modular on the lower product spaee $R\times S$ , which will be call ed the lower modular.A moclular $m$ on $a$ opwmel product $s\varphi au$ $W$ is a product modular, if and only if $m(z)\leqq m_{1}(z)$ for $z\in R\times S$ , and its $c\sigma ngugate$ modular is a product modular fron $\overline{R}$ and $\overline{S}$ , if and only if $ m(z)\geqq$ $m_{u}(\grave{z})f\sigma rz\in W$ .In this paper we shall make.use of the notations in the book: H. NAKANO, Modulared semi-ordered linear spaces, Tokyo (1950).This book will be denoted by MSLS. \S 1. Bounded bilinear functionalsLet $R$ and $S$ be two lattice ordered linear spaces.A functional $\varphi(x, y)(x\in R, y\in S)$ is said to be bilinear; if $\varphi(x_{1}+x_{2}, y)=\varphi(x_{1},..y)+\varphi(x_{-},.y)$, $\varphi(x, y_{1}-\vdash y_{d},)=\varphi(x, y_{1},)+\varphi(x, y_{\lrcorner}?)$ , $\varphi(ax, y)=\varphi(x, (ay)=\alpha\varphi(x, y)