Order convergence in infinite-dimensional vector lattices is not topological

In this note, we show that the order convergence in a vector lattice $X$ is not topological unless $\dim X<\infty$. Furthermore, we show that, in atomic order continuous Banach lattices, the order convergence is topological on order intervals

Unbounded p-Convergence in Lattice-Normed Vector Lattices

A net xα in a lattice-normed vector lattice (X, p, E) is unbounded p-convergent to x ∈ X if p(| xα− x| ∧ u) → o 0 for every u ∈ X+. This convergence has been investigated recently for (X, p, E) = (X, |·|, X) under the name of uo-convergence, for (X, p, E) = (X, ‖·‖, ℝ) under the name of un-convergence, and also for (X, p, ℝX ′) , where p(x)[f]:= |f|(|x|), under the name uaw-convergence. In this paper we study general properties of the unbounded p-convergence.Article Pre-prin

Internal characterization of Brezis-Lieb spaces

In order to find an extension of Brezis-Lieb's lemma to the case of nets, we replace the almost everywhere convergence by the unbounded order convergence and introduce the pre-Brezis-Lieb property in normed lattices. Then we identify a wide class of Banach lattices in which the Brezis-Lieb lemma holds true. Among other things, it gives an extension of the Brezis-Lieb lemma for nets in L-p for p is an element of[1,infinity)

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

Unbounded p-Convergence in Lattice-Normed Vector Lattices

A net x? in a lattice-normed vector lattice (X, p, E) is unbounded p-convergent to x ? X if p(| x?? x| ? u) ? o 0 for every u ? X+. This convergence has been investigated recently for (X, p, E) = (X, |·|, X) under the name of uo-convergence, for (X, p, E) = (X, ?·?, ?) under the name of un-convergence, and also for (X, p, ?X ?) , where p(x)[f]:= |f|(|x|), under the name uaw-convergence. In this paper we study general properties of the unbounded p-convergence. © 2019, Allerton Press, Inc

Compact-like operators in lattice-nonmed. spaces

WOS: 000429511400008A linear operator T between two lattice-normed spaces is said to be p-compact if, for any p-bounded net x(alpha),,the net Tx(alpha) has a p-convergent subnet. p-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, AM-compact operators, etc. Similar to M-weakly and L-weakly compact operatois, we define p-M-weakly and p-L-weakly compact operators and study some of their properties. We also study up-continuous and up"compact operators between lattice nonmed vector lattices. (C) 2017 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved