Topological vector lattice
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In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order 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.
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Definition
If is a vector lattice then by the vector lattice operations we mean the following maps:
- the three maps to itself defined by , , , and
- the two maps from into defined by and.
If is a TVS over the reals and a vector lattice, then 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 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 is a topological vector space (TVS) and an ordered vector space then is called locally solid if possesses a neighborhood base at the origin consisting of solid sets.[1] A topological vector lattice is a Hausdorff TVS 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 denote the set of all bounded subsets of a topological vector lattice with positive cone and for any subset , let be the -saturated hull of . Then the topological vector lattice's positive cone is a strict -cone,[1] where is a strict -cone means that is a fundamental subfamily of that is, every is contained as a subset of some element of ).[2]
If a topological vector lattice is order complete then every band is closed in .[1]
Examples
The Lp spaces () are Banach lattices under their canonical orderings. These spaces are order complete for .
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
Bibliography
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