Continuous lattice ordering by Schauder basis cones

Abstract

Let (E, τ \tau ) be a barrelled Hausdorff space lattice ordered by the cone of an unconditional Schauder basis ( x n , f n ) ({x_n},{f_n}) . It is shown that under such an ordering (E, T) is a locally convex lattice. Necessary and sufficient conditions are given for the lattice operations to be continuous with respect to the weak topologies on E and its topological dual E ′ E’ : the lattice operations are σ ( E , E ′ ) \sigma (E,E’) -continuous on E if and only if { f n : n ∈ ω } \{ {f_n}:n \in \omega \} is a Hamel basis for E ′ E’ .