Topological vector lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) $$X$$ that has a partial order $$\,\leq\,$$ 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.

Definition
If $$X$$ is a vector lattice then by the vector lattice operations we mean the following maps: If $$X$$ is a TVS over the reals and a vector lattice, then $$X$$ is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.
 * 1) the three maps $$X$$ to itself defined by $$x \mapsto|x |$$, $$x \mapsto x^+$$, $$x \mapsto x^{-}$$, and
 * 2) the two maps from $$X \times X$$ into $$X$$ defined by $$(x, y) \mapsto \sup_{} \{ x, y \}$$ and$$(x, y) \mapsto \inf_{} \{ x, y \}$$.

If $$X$$ 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 $$X$$ is a topological vector space (TVS) and an ordered vector space then $$X$$ is called locally solid if $$X$$ possesses a neighborhood base at the origin consisting of solid sets. A topological vector lattice is a Hausdorff TVS $$X$$ that has a partial order $$\,\leq\,$$ 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 $$\mathcal{B}$$ denote the set of all bounded subsets of a topological vector lattice with positive cone $$C$$ and for any subset $$S$$, let $$[S]_C := (S + C) \cap (S - C)$$ be the $C$-saturated hull of $$S$$. Then the topological vector lattice's positive cone $$C$$ is a strict $$\mathcal{B}$$-cone, where $$C$$ is a strict $$\mathcal{B}$$-cone means that $$\left\{ [B]_C : B \in \mathcal{B} \right\}$$ is a fundamental subfamily of $$\mathcal{B}$$ that is, every $$B \in \mathcal{B}$$ is contained as a subset of some element of $$\left\{ [B]_C : B \in \mathcal{B} \right\}$$).

If a topological vector lattice $$X$$ is order complete then every band is closed in $$X$$.

Examples
The Lp spaces ($$1 \leq p \leq \infty$$) are Banach lattices under their canonical orderings. These spaces are order complete for $$p < \infty$$.