Topological vector lattice

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

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

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.

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.

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. 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.

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space. 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, 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 ${\displaystyle \left\{[B]_{C}:B\in {\mathcal {B}}\right\}}$).

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

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