Vol. 37, No. 1, 1971

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Strictly increasing Riesz norms

Lawrence Carlton Moore

Vol. 37 (1971), No. 1, 171–180

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
Received: 15 July 1970
Published: 1 April 1971
Lawrence Carlton Moore