Unbounded order convergence in dual spaces

By Niushan Gao. 2013

71 citations recorded by the source corpus.

A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_\alpha-x}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. We also characterize among $\sigma$-order complete Banach lattices the spaces in whose dual space every simultaneously uo- and $w^*$-convergent sequence converges weakly/in norm.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/20d405fea61230fc0f35e8e0bae98507a964b6c3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. O. Zabeti
2. V. G. Troitsky
3. M. Kandić
4. M. Marabeh
5. Foivos Xanthos

In Mathematical Physics

1. O. Zabeti
2. V. G. Troitsky
3. M. Kandić
4. M. Marabeh
5. Foivos Xanthos
6. Niushan Gao
7. M. O’Brien
8. Y. Deng

In Statistics and Probability

1. O. Zabeti
2. V. G. Troitsky
3. Foivos Xanthos
4. M. Kandić
5. M. Marabeh
6. Niushan Gao
7. M. O’Brien
8. Y. Deng

Citations in the corpus

• Uo-convergence and its applications to Cesàro means in Banach lattices (2015, methodology): “We refer to [26, 27] for some basic properties of uo-convergence and uo-Cauchy.”
• Unbounded norm convergence in Banach lattices (2016, background): “First, while [GTX, Corollary 3.6] asserts that every disjoint sequence is uonull (see also [Gao14, Lemma 1.1]), this example shows that a disjoint sequence need not be un-null.”
• Unbounded norm topology in Banach lattices (2016, methodology): “Propositions 4.3, 4.4, and 4.6 in [Gao14] remain valid under the additional assumption thatX∗ is order continuous (note that the dual positive Schur property already implies that X∗ is order continuous by [Wnuk13, Proposition 2.1]).”
• Unbounded absolute weak convergence in Banach lattices (2016, result): “nsider E = C([0,1]). Define the sequence (fn) ⊆ E via 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 b”