Analysis Seminar

Niushan Ga, Southwest Jiaotong University

“Unbounded Order Convergence in Banach lattices”

A net (xα) in a vector lattice X unbounded order converges to 0 if xα ∧ y →−o 0 in X for any uo

uo a.e.
In the case where X is a Banach function space, it can be shown that xn −→ 0 iff xn −−→ 0,

o a.e.
while xn →− 0 iff xn −−→ 0 and xn ≤ F for some F ∈ X and all n ≥ 1. The last condition

means that the sequence (xn) is order bounded, i.e. it is contained in an order interval [−F,F]. As this condition is generally difficult to satisfy, it suggests that unbounded order convergence is more useful than order convergence. In this talk, we discuss some fundamental properties of unbounded order convergence and also some applications of it. In particular, we use it to show that every Banach lattice has at most one (up to lattice isometries) order continuous predual.

MC 5403 **Please Note Room **