Towards a theory of unbounded locally solid Riesz spaces

Abstract

We introduce the notion of unbounded locally solid Riesz spaces, and investigate its fundamental properties.

On Some Aspects of Pseudonorm Compact and Montel Operators on Locally Solid Vector Lattices

“…In this case, we write x α un −→ x. A systematic study of unbounded convergences can be found in [5,7,9,15,16,18,24]. In general, unbounded order convergence is not topological.…”

“…A locally solid topology on a vector lattice X can be used to construct a new unbounded locally solid topology on X. We refer the reader to [4,6,7,9] for more details. Let X be a vector lattice.…”

“…Comparison of various versions of boundedness forms the basic method of our analyses throughout. Most of our results are motivated from [9,4,7]. Recent manuscripts, see [6,9,15,16,18,5,24,10,27], about various types of convergences on vector lattices, and, the recent progress towards the operator theory on both vector lattice valued pseudonormed spaces and on spaces equipped with unbounded convergences, see [4,6,7,10,11,12,22,23] and the references therein, stimulated us to write this article.…”

“…Most of our results are motivated from [9,4,7]. Recent manuscripts, see [6,9,15,16,18,5,24,10,27], about various types of convergences on vector lattices, and, the recent progress towards the operator theory on both vector lattice valued pseudonormed spaces and on spaces equipped with unbounded convergences, see [4,6,7,10,11,12,22,23] and the references therein, stimulated us to write this article. Some of the present material can also be profitably used in the settings of vector lattice valued pseudonormed spaces.…”

“…For instance, Fatou and order continuous pseudonorms are among the most useful pseudonorms. In addition to the fact that main results of [9] establish the relationships between unbounded convergences and pseudonorms on vector lattices, results presented in Section 3 abstract further properties of these special pseudonorms which can be combined with the settings of [6]. In Section 4, we study compactness properties of operators between two pseudonormed vector lattices.…”