CONVEXITY AND EVENNESS IN MODULARED SEMI-ORDERED LINEAR SPACES
Tsuyoshi Andô · Hokkaido Mathematical Journal · 1959
In the remainder of this Introduction, we shall state definitions and results used later from the theory of modulared semi-ordered linear spaces 2) For Orlicz spaces, see [17].Convexity and Evenness in Modulared Semi-ordered Linear Spaces 61 developed by H. Nakano in $[12, 13]$ .$\tilde{R}$ and $\overline{R}$ denote the totality of all linear functionals and that of all universally continuous linear functionals on $R$ respectively, which are bounded under the norm.On $\tilde{R}$ the associated modular $\tilde{m}$ of $m$ is de- fined by the formula:$\tilde{m}$ satisfies all the modular conditions (see [12; \S 38]), and ' (11) $\tilde{a}(a)\leqq m(a)+\tilde{m}(\tilde{a})$for all $a\in R,\tilde{a}\epsilon\tilde{R}$ .When we consider the associated modular only on $\overline{R}$ , we call it the conjugate modular of $m$ and denote it by $\overline{m}$ .In this paper projectors $[p]$ and projection operators $[N]$ are frequently used (for the definition seeIn this paper we always assume semi-regularity of $R$ , i.e. for any $0 eq a\in R$ there exists $\overline{a}\in\overline{R}$ such that $\overline{a}(a) eq 0$ .By semi-regularity the following formulas are valid (see [11], [12; \S 39-40] and [13; \S 83])$m(a)=u\frac{s}{x}\in\frac{p}{R}\{\overline{x}(a)-\overline{m}(\overline{x})\}$ for all $a\in R$ ;(13) 11 $a||=\sup_{\overline{m}(\overline{x})\leqq 1}|\overline{x}(a)|$ for all $a\in R'$ .Two norms satisfy always (see [12; \S 40])111 $a|||\leqq||a||\leqq 2|||a$ III for all $a\in^{t}R\backslash $ ,