Unbounded norm topology in Banach lattices

A net (x(alpha)) in a Banach lattice X is said to un-converge to a vector x if xl A parallel to vertical bar x(alpha) - x vertical bar boolean AND u parallel to -> 0 for every u is an element of X+. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quaRi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball B-x is un-complete. For a Banach lattice X, B-x is un-compact if X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence. (C) 2017 Elsevier Inc. All rights reserved.