ON THE CONTINUITY AND THE MONOTONOUS, NESS OF NORMS By

\S 1. Let $R $ be a universally continuous semi-ordered linear space1) ( $i.e $. conditionally complete vector lattice in Birkhoff’s sense) and $|| $. I be a norm on $R $ satisfying the following conditions throughout this paper: (N. 1) $|x|\leqq|y|(x, y\in R) $ implies $||x||\leqq||y|| $; (N. 2) $0\leqq x_{\lambda}\uparrow_{\lambda\in\Lambda}x $ implies $||x||=\sup_{\lambda\in A}||x_{\lambda}||^{2)} $. A norm $||\cdot|| $ on $R $ is, called continuous, if (1.1) $\inf_{\nu=1,2}\ldots||x_{\nu}||=0 $ for any $x_{\nu}\downarrow_{\nu=1}^{\infty}0^{3)} $. The continuity of norms on $R $ plays an important r\^ole in the theory of semi-ordered linear spaces. In fact, it is well known [8, 9; \S 31] that every norm-continuous hnear functional $f $ on $\dot{R} $ is (order-) universally continuous, $i.e $. (1.2) $\inf_{\lambda\Lambda}|f(x_{\lambda})|=0 $ for any $x_{\lambda}\downarrow_{\lambda\in A}0 $, and $R $ becomes superuniversally continuous4) as a space in this case. It is clear that if a norm $|| $. I on $R $ is continuous, the another norm $||\cdot||_{1} $ which is equivalent to $||\cdot|| $ is also continuous. As for the conditions under which norms $||\cdot|| $ on $R $ are continuous, there are the detailed in-vestigations by T. And\^o $[3, 4] $. A norm $|| $. I on $R $ is called monotone [8], if (1.3) $|x|\leqq|y|(x, y\in R) $ implies $||x||\neq^{\sim}||y|| $, and is called uniformly monotone [8, 9; \S 30], if 1) This terminology is due to H. Nakano [9]. We use mainly notation and terminology of [9] here. 2) A norm satisfying (N. 1) and (N. 2) is called semi-continuous in [10]. A norm on $||\cdot|| $ satisfying (N. 1) is called monotone in [7]. On the other hand, (N. 1) is assumed for any norm of normed lattices in [6]. 3) This means $x_{1}\geqq x_{2}\geqq\cdots\geqq 0 $ and $\infty\bigcap_{\nu=1}x_{\nu}=0 $. 4) $R $ is calle