In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) ${\displaystyle X}$ that has a partial order ${\displaystyle \,\leq \,}$ making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.^[1] Ordered vector lattices have important applications in spectral theory.

Definition

If ${\displaystyle X}$ is a vector lattice then by the vector lattice operations we mean the following maps:

1. the three maps ${\displaystyle X}$ to itself defined by ${\displaystyle x\mapsto |x|}$, ${\displaystyle x\mapsto x^{+))$, ${\displaystyle x\mapsto x^{-))$, and
2. the two maps from ${\displaystyle X\times X}$ into ${\displaystyle X}$ defined by ${\displaystyle (x,y)\mapsto \sup _{}\{x,y\))$ and${\displaystyle (x,y)\mapsto \inf _{}\{x,y\))$.

If ${\displaystyle X}$ is a TVS over the reals and a vector lattice, then ${\displaystyle X}$ is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.^[1]

If ${\displaystyle X}$ 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.^[1]

If ${\displaystyle X}$ is a topological vector space (TVS) and an ordered vector space then ${\displaystyle X}$ is called locally solid if ${\displaystyle X}$ possesses a neighborhood base at the origin consisting of solid sets.^[1] A topological vector lattice is a Hausdorff TVS ${\displaystyle X}$ that has a partial order ${\displaystyle \,\leq \,}$ making it into vector lattice that is locally solid.^[1]

Properties

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.^[1] Let ${\displaystyle {\mathcal {B))}$ denote the set of all bounded subsets of a topological vector lattice with positive cone ${\displaystyle C}$ and for any subset ${\displaystyle S}$, let ${\displaystyle [S]_{C}:=(S+C)\cap (S-C)}$ be the ${\displaystyle C}$-saturated hull of ${\displaystyle S}$. Then the topological vector lattice's positive cone ${\displaystyle C}$ is a strict ${\displaystyle {\mathcal {B))}$-cone,^[1] where ${\displaystyle C}$ is a strict ${\displaystyle {\mathcal {B))}$-cone means that ${\displaystyle \left\{[B]_{C}:B\in {\mathcal {B))\right\))$ is a fundamental subfamily of ${\displaystyle {\mathcal {B))}$ that is, every ${\displaystyle B\in {\mathcal {B))}$ is contained as a subset of some element of ${\displaystyle \left\{[B]_{C}:B\in {\mathcal {B))\right\))$).^[2]

If a topological vector lattice ${\displaystyle X}$ is order complete then every band is closed in ${\displaystyle X}$.^[1]

Examples

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