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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 is possesses a neighborhood base at the origin consisting of solid sets.[1] Ordered vector lattices have important applications in spectral theory.

Definition

If is a vector lattice then by the vector lattice operations we mean the following maps:

  1. the three maps to itself defined by , , , and
  2. 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

References

  1. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 234–242.
  2. ^ Schaefer & Wolff 1999, pp. 215–222.

Bibliography