On the norms by uniformly finite modulars
Tetsuya Shimogaki · Proceedings of the Japan Academy Series A Mathematical Sciences · 1957
Let R be a modulared semi-ordered linear space and m be a modular on R. On R we can define two norms as follows:Ix]! is said to be the first norm by m and Ill'Ill is said to be the second norm or the modular norm by m.Since we have !11!11!1 !i 2 Ill Ill for every R (cf. 4),they are equivalent to each other.