Topological vector lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory. If is a vector lattice then by the vector lattice operations we mean the following maps: the three maps to itself defined by , , , and the two maps from into defined by and. If is a TVS over the reals and a vector lattice, then is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous. If is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous. If is a topological vector space (TVS) and an ordered vector space then is called locally solid if possesses a neighborhood base at the origin consisting of solid sets. A topological vector lattice is a Hausdorff TVS that has a partial order making it into vector lattice that is locally solid. Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space. Let denote the set of all bounded subsets of a topological vector lattice with positive cone and for any subset , let be the -saturated hull of . Then the topological vector lattice's positive cone is a strict -cone, where is a strict -cone means that is a fundamental subfamily of that is, every is contained as a subset of some element of ). If a topological vector lattice is order complete then every band is closed in . The Banach spaces () are Banach lattices under their canonical orderings. These spaces are order complete for .