Unbounded Norm Topology in Banach Lattices
نویسنده
چکیده
A net (xα) in a Banach lattice X is said to un-converge to a vector x if ∥∥|xα−x|∧u∥∥→ 0 for every u ∈ 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 quasi-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 BX is un-complete. For a Banach lattice X, BX is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.
منابع مشابه
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