Topological Vector Lattice

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) [math]\displaystyle{ X }[/math] that has a partial order [math]\displaystyle{ \,\leq\, }[/math] 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 [math]\displaystyle{ X }[/math] is a vector lattice then by the vector lattice operations we mean the following maps:

1. the three maps [math]\displaystyle{ X }[/math] to itself defined by [math]\displaystyle{ x \mapsto|x | }[/math], [math]\displaystyle{ x \mapsto x^+ }[/math], [math]\displaystyle{ x \mapsto x^{-} }[/math], and
2. the two maps from [math]\displaystyle{ X \times X }[/math] into [math]\displaystyle{ X }[/math] defined by [math]\displaystyle{ (x, y) \mapsto \sup_{} \{ x, y \} }[/math] and[math]\displaystyle{ (x, y) \mapsto \inf_{} \{ x, y \} }[/math].

If [math]\displaystyle{ X }[/math] is a TVS over the reals and a vector lattice, then [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] is a topological vector space (TVS) and an ordered vector space then [math]\displaystyle{ X }[/math] is called locally solid if [math]\displaystyle{ X }[/math] possesses a neighborhood base at the origin consisting of solid sets.^[1] A topological vector lattice is a Hausdorff TVS [math]\displaystyle{ X }[/math] that has a partial order [math]\displaystyle{ \,\leq\, }[/math] 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 [math]\displaystyle{ \mathcal{B} }[/math] denote the set of all bounded subsets of a topological vector lattice with positive cone [math]\displaystyle{ C }[/math] and for any subset [math]\displaystyle{ S }[/math], let [math]\displaystyle{ [S]_C := (S + C) \cap (S - C) }[/math] be the [math]\displaystyle{ C }[/math]-saturated hull of [math]\displaystyle{ S }[/math]. Then the topological vector lattice's positive cone [math]\displaystyle{ C }[/math] is a strict [math]\displaystyle{ \mathcal{B} }[/math]-cone,^[1] where [math]\displaystyle{ C }[/math] is a strict [math]\displaystyle{ \mathcal{B} }[/math]-cone means that [math]\displaystyle{ \left\{ [B]_C : B \in \mathcal{B} \right\} }[/math] is a fundamental subfamily of [math]\displaystyle{ \mathcal{B} }[/math] that is, every [math]\displaystyle{ B \in \mathcal{B} }[/math] is contained as a subset of some element of [math]\displaystyle{ \left\{ [B]_C : B \in \mathcal{B} \right\} }[/math]).^[2]

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

Examples

The Banach spaces [math]\displaystyle{ L^p(\mu) }[/math] ([math]\displaystyle{ 1 \leq p \leq \infty }[/math]) are Banach lattices under their canonical orderings. These spaces are order complete for [math]\displaystyle{ p \lt \infty }[/math].

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Complemented lattice
• Fréchet 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

References

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. 

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