Strictly increasing Riesz norms

Lawrence Carlton Moore

Vol. 37 (1971), No. 1, 171–180

DOI: 10.2140/pjm.1971.37.171

Abstract

Let L be a Riesz space and ρ a Riesz norm on L. Then ρ is said to be strictly increasing if u,v ∈ L and 0 ≦ u≦̸v imply that ρ(u) < ρ(v). We investigate necessary conditions and sufficient conditions that for a given Riesz norm there is an equivalent strictly increasing Riesz norm. A necessary condition is that the Riesz space possess the countable sup property. A sufficient condition is that the given norm be an (A,ii) norm. Finally, we investigate the relationship between the existence of strictly increasing Riesz norms and the Souslin hypothesis.

Mathematical Subject Classification 2000

Primary: 46A40

Secondary: 46B05, 04A15

Milestones

Received: 15 July 1970

Published: 1 April 1971