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

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Complemented lattice
• Fréchet lattice – Topological vector lattice
• Locally convex vector lattice
• Normed lattice
• Ordered vector space – Vector space with a partial order
• Pseudocomplement
• Riesz space – Partially ordered vector space, ordered as a lattice

References

1. Schaefer & Wolff 1999, pp. 234–242.
2. Schaefer & Wolff 1999, pp. 215–222.

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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.