Topological vector lattice explained

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 possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory.

Definition

If

is a vector lattice then by the vector lattice operations we mean the following maps:

1. the three maps

to itself defined by , , , and

1. 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.

Properties

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

C

-saturated hull of . Then the topological vector lattice's positive cone is a strict -cone, where is a strict

l{B}

-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 .

Examples

The L^p spaces (

) are Banach lattices under their canonical orderings. These spaces are order complete for