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Concept
Topological vector lattice
Summary
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 is possesses a neighborhood base at the origin consisting of solid sets.
Ordered vector lattices have important applications in spectral theory.
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.
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.
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.
A topological vector lattice is a Hausdorff TVS that has a partial order making it into vector lattice that is locally solid.
Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.
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, where is a strict -cone means that is a fundamental subfamily of that is, every is contained as a subset of some element of ).
If a topological vector lattice is order complete then every band is closed in .
The Banach spaces () are Banach lattices under their canonical orderings.
These spaces are order complete for .
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