Topological vector lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) ${\displaystyle X}${\displaystyle X} that has a partial order ${\displaystyle ,円\leq ,円}${\displaystyle ,円\leq ,円} 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.

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

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

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

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

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

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

• 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
• Normed lattice
• Ordered vector space – Vector space with a partial order
• Pseudocomplement
• Riesz space – Partially ordered vector space, ordered as a lattice

• 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.

• Functional analysis

• Articles needing additional references from June 2020
• All articles needing additional references