A closed graph theorem for order bounded operators. Issue 2 (31st March 2016)

• Record Type: Journal Article
• Authors: van der Walt, Jan Harm
• Is Part Of: Quaestiones mathematicae. Volume 39:Issue 2(2016)
• Page Start: 167
• Page End: 178
• Publication Date: 2016-03-31
• Subjects: Primary 46A40 -- 47B65; -- Secondary 46A19; Vector lattice -- convergence vector space -- order bounded operator; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2015.1068235
• Languages: English
• ISSNs: 1607-3606

Abstract

Abstract: The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a difficult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.