Vol. 71, No. 1, 1977

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A comparison of the relative uniform topology and the norm topology in a normed Riesz space

Lawrence Carlton Moore

Vol. 71 (1977), No. 1, 107–118

It is well-known that if a normed Riesz space (L,ρ) is ρ-complete, i.e., a Banach lattice, then the relative uniform topology and the norm topology are the same. Under weaker conditions the two topologies show some measure of agreement. In particular if L has (i) the local σ-property (a property weaker than both the σ-property and local completeness) and (ii) the property that for every ideal A the norm closure of A equals the set of limit points of relatively uniformly convergent sequences of elements of A, then every sequence un 0 with ρ(un) 0 is a relatively uniformly convergent sequence. (This generalizes a theorem of Luxemburg and Zaanen.) However conditions (i) and (ii) are not sufficient to imply that the relative uniform topology and the norm topology agree on order intervals. Examples are given illustrating increasing degrees of agreement of the two topologies.

Mathematical Subject Classification 2000
Primary: 46A40
Received: 8 October 1976
Published: 1 July 1977
Lawrence Carlton Moore