ON AN EQUIVALENCE RELATION ON SEMI-ORDERED LINEAR SPACFS
Tetsuya Shimogaki · Hokkaido Mathematical Journal · 1964
A Banach function space $X$ is said to have the strong rearrangement in- variant property (s-RIP), if $f\in X,$ $f\sim q$ implies $g\in X$ and $\Vert g\Vert\leqq A\Vert f\Vert$ , where 5,6]$ and I. Halperin 1) A semi-ordered linear space $R$ is called universally continuous, if $0\leqq a_{\lambda}(\lambda\in\Lambda)$ implies $\bigcap_{\lambda\epsilon A}a_{\lambda}\in R,$ $i.e$ .a conditionally complete vector lattice in Birkhoff's sense or a K.space in the sense of Vulich [12].2) For the detailed properties of Banach function spaces see [7] or [13].