Applications of convergence spaces to vector lattice theory (Q2850630)

The concept of a locally solid convergence vector lattice (LSCVL, for short) is introduced as a generalization of locally solid Riesz spaces. In general, neither initial nor final constructions preserve local solidity. However, it is shown that it is possible to form special initial and final convergence structures in the class of LSCVL's which, in general, do not coincide with the initial and final vector space convergence structures. Sufficient conditions are given for the initial convergence structure to be locally solid.NEWLINENEWLINEThere are two dual structures for an LSCVL -- the topological dual and the order dual, the first one being of more importance. Namely, the author shows that the topological dual of an LSCVL can be equipped with the convergence vector structure and it becomes a locally solid vector lattice. This implies that every complete, Hausdorff locally convex, locally solid Riesz space is isomorphic, both as a convergence vector space and as a vector lattice, to its second dual. Completeness and completion of LSCVL's are also described.NEWLINENEWLINEFinally, as applications, a closed graph theorem for a class of LSCVL's is proved and a duality result for locally convex, locally solid Riesz spaces is presented.NEWLINENEWLINESeveral examples are given throughout the paper. As a result, the notion of LSCVL provides an appropriate context for a number of natural modes of convergence that cannot be described in the usual topological terms.