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 ≤ making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.[1] 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 ${\displaystyle x\mapsto |x|}$, ${\displaystyle x\mapsto x^{+}}$, ${\displaystyle x\mapsto x^{-}}$, and
2. the two maps from ${\displaystyle X\times X}$ into X defined by ${\displaystyle (x,y)\mapsto \sup \left\{x,y\right\}}$ and${\displaystyle (x,y)\mapsto \inf \left\{x,y\right\}}$.

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.[1]

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.[1]

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.[1] A topological vector lattice is a Hausdorff TVS X that has a partial order ≤ 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 C and for any subset S, let ${\displaystyle [S]_{C}:=\left(S+C\right)\cap \left(S-C\right)}$ be the C-saturated hull of S. Then the topological vector lattice's positive cone C is a strict ${\displaystyle {\mathcal {B}}}$-cone,[1] where C is a strict ${\displaystyle {\mathcal {B}}}$-cone means that ${\displaystyle \left\{\left[B\right]_{C}:B\in {\mathcal {B}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {B}}}$ (i.e. every ${\displaystyle B\in {\mathcal {B}}}$ is contained as a subset of some element of ${\displaystyle \left\{\left[B\right]_{C}:B\in {\mathcal {B}}\right\}}$).[2]

If a topological vector lattice X is order complete then every band is closed in X.[1]

Examples

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

See also

• Banach lattice
• Fréchet lattice
• Locally convex vector lattice
• Normed lattice
• Ordered vector space
• Riesz space

References

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

• 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.