Let $R $ be a universally continuous semi-ordered linear space. A functional $m(a)(a\in R) $ is said to be a modularl) on $R $ if it satisfies the following modular conditions: (1) $ 0\leqq m(a)\leqq\infty $ for all $a\in R $; (2) if $m(\xi a)=0 $ for all $\xi>0 $ , then $a=0 $; (3) for any $a\in R $ there exists $a>0 $ such that $ m(aa)<\infty $; (4) for every $a\in R, $ $m(\xi a) $ is a convex function of $\xi $; (5) $|a|\leqq|b| $ implies $m(a)\leqq m(b) $; (6) $a\wedge b=\backslash 0 $ implies $m(a+b)=m(a)+m(b) $; (7) $0\leqq a_{l}\uparrow aRC-.4 $ implies $m(a)=\sup_{R\in\Lambda}m(a_{\lambda}) $. In $R $ , we define functionals $||a||, $ $\Vert|a\Vert|(a\in R) $ as follows $||a||=\inf_{\xi>0}\frac{1+m(\tilde{\sigma}a)}{\xi} $. $\Vert|a|_{1}^{1}|=\inf_{m(\text{\’{e}} a)\leq 1}\frac{1}{|\xi|} $. Then it is easily seen that both $||a|| $ and $\Vert|a\Vert| $ are norms on $R $ and $||_{1}|a\Vert|\leqq||a||\leqq 2\Vert|a\Vert| $ for all $a\in R $. $||a|| $ is said to be the first norm by $m$ and $\Vert|a\Vert| $ is said to be the second norm by $m $. Let $\overline{R}^{m} $ be the modular

Let $R $ be a universally continuous semi-ordered linear space. A functional $m(a)(a\in R) $ is said to be a modular on $R $ if it satisfies the following modular conditions: (1) $ 0\leqq m(a)\leqq+\infty $ for all $a\in R $; (2) if $m(\hat{\sigma}a)=0 $ for all $\xi\geqq 0 $ , then $a=0 $; (3) for any $a\in R $ there exists $a>0 $ such that $ m(aa)<+\infty $; (4) for every $a\in R, $ $m(\xi a) $ is a convex function of $\xi $; (5) $|a|\leqq|b| $ implies $m(a)\leqq m(b) $; (6) $a\leftrightarrow b=0 $ implies $m(a+b)=m(a)+m(b) $; (7) $0\leqq a_{l}\uparrow_{l\in\Lambda}a_{\backslash} $ implies $m(a)=\sup_{\lambda\in\Lambda}m(a_{\lambda}) $. Throughout the paper we use the notations and terminologies used in [2]. Here $|w|-\lim_{\nu\rightarrow\infty}a_{\nu}=a $ or $w-\lim_{\nu\rightarrow\infty}a_{\nu}=a $ for $a, $ $a_{\nu}\in R(\nu=1,2,3, \cdots)$ means $\lim_{\nu\rightarrow\infty}|\overline{\sigma}|(|a_{\nu}-a|)=0 $ or $\lim_{\nu\rightarrow\infty} $ a $(a_{\nu}-a)=0 $ respectively for any $\overline{a}\in\overline{R}^{m1)} $. If $\Phi(u) $ is a real convex function, defined for $u\geqq 0 $ , such that $\Phi(0)=0$ and $\Phi(u)\geqq 0 $ for $u>0 $ , but $\Phi(u) $ not identically zero or infinity for $u>0 $,

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