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.
