turkmath.org

A net $(X_\alpha)$ in a Banach lattice is said to be unbounded norm convergent or un-convergent to if $\||x_\alpha-x|\wedge u\|$ for all $u\in X_+$ . In this talk, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We will see that un- topology agrees with the norm topology iff $X$ has a strong unit. Un-topology is metrizable iff $X$ has a quasi-interior point. Suppose that is order continuous, then un-topology is locally convex iff 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 iff $X$ is an atomic KB-space.