Unbounded norm topology in Banach lattices
By M. Kandić, M. Marabeh, V. G. Troitsky. 2016
90 citations recorded by the source corpus.
Abstract A net ( x α ) in a Banach lattice X is said to un-converge to a vector x if ‖ | x α − 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 B X is un-complete. For a Banach lattice X, B X 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.
Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c260e464ca06ed6dbf0354a6dffc9e395135c8e1
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In Geometry and Topology
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Citations in the corpus
- Unbounded absolute weak convergence in Banach lattices (2016, background): “fn(0) = 1, fn(1 n ) = fn(1) = 0, and linear in between. Then (fn) is uo-null but not uaw-null. The following result is analogous to Theorem 2.1 in [G14]; the proof is similar (cf. also Theorem 8.1 in [KMT17]). Note that sequences may be replaced by nets. Proposition 5. The following are equivalent: ”