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 $\le$ making it into vector lattice that possesses a neighborhood base at the origin consisting of solid sets.Template:Sfn 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:

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 $(x,y)\mapsto \underset{}{\sup }\{x,y\}$ and$(x,y)\mapsto \underset{}{\inf }\{x,y\}$.

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.Template:Sfn

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.Template:Sfn

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.Template:Sfn A topological vector lattice is a Hausdorff TVS $X$ that has a partial order $\le$ making it into vector lattice that is locally solid.Template:Sfn

Properties

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.Template:Sfn Let $ℬ$ 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 $ℬ$-cone,Template:Sfn where $C$ is a strict $ℬ$-cone means that $\{[B{]}_C:B\in ℬ\}$ is a fundamental subfamily of $ℬ$ that is, every $B\in ℬ$ is contained as a subset of some element of $\{[B{]}_C:B\in ℬ\}$).Template:Sfn

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

Examples

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

See also

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Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces