Topological vector lattice

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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Related concepts (2)

Ordered vector space

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Given a vector space over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied implies implies If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure.

Riesz space

In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem.