Topological vector lattice

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

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

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

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

Examples

The L^p spaces ( 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } ) are Banach lattices under their canonical orderings. These spaces are order complete for p < ∞ {\displaystyle p<\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
• Normed 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.