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 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:

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.

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 $\le$ 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 ${ℬ}$ 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, 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 $\{[B{]}_C:B\in {ℬ}\}$).

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

Examples

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

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Complemented lattice – Bound lattice in which every element has a complement
• Fréchet lattice – Topological vector lattice
• Locally convex vector lattice
• Ordered vector space – Vector space with a partial order
• Pseudocomplement
• Riesz space – Partially ordered vector space, ordered as a lattice

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.