Concave modulars.
Hidegorô Nakano · Journal of the Mathematical Society of Japan · 1953
We have defined and discussed modulars on semi.orderedlinear space in a book1).Let $R$ be a semi-ordered linear space and univer- sally continuous, that is, for every system of positive elements $a_{\lambda}\in R$ $(\lambda\in\Lambda)$In this paper we shall consider a functional $m(x)(x\in R)$ which satisfies instead of 4) the condition: $m(\xi x)$ is a concave function of $\xi\geq 0$ , i.e., we define a concave modular $m(x)(x\in R)$ by the postu.implies the existence of an element $x_{0}$ for which $x_{ u}\uparrow_{ u\approx I}^{\infty}x_{0}$ and $m(x_{0})=\lim_{ u\backslash \rightarrow\infty}m(x_{ u})$ .Concerning the concave modulars $m(x)$ on $R$ , we can prove $m(x+y)\leqq m(x)+m(y)$ for every $x,y\in R$ . Thus, every concave modularand hence there exists the limit $ m_{1}(x)=\lim_{\xi\rightarrow\infty}\underline{m}(\xi x)\xi$ for every $x\in R$ .A concave modular $m(x)$ is said t\'o be of the first kind, if $m_{1}(x)=0$ implies $x=0$, and of the second kind, if $m_{1}(x)=0$ for every $x\in R$ .With this definition, $R$ may be devided in two normal manifolds $F$ and $S$ such that $m(x)$ is of the first kind in $F$ and of the second kind in $S$ .If $m(x)$ is of the first kind in $R$ , then $m_{1}(x)$ is a norm on $R$ and $m_{1}(x+y)=m_{1}(x)+m_{1}(y)$ for $x,y\geqq 0$ .By this norm $m_{1}(x),$ $R$ is complete, and hence a so.called generalized $L_{1}\cdot space$ , if and only $if\sup_{m_{1}(x)\leq 1}m(x)<+\infty$ .