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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Concepts associés (2)

Espace vectoriel ordonné

En mathématiques, un espace vectoriel ordonné (ou espace vectoriel partiellement ordonné) est un espace vectoriel sur muni d'une relation d'ordre compatible avec sa structure. Il est dit totalement ordonné si l'ordre associé est un ordre total. Soit E un espace vectoriel sur le corps des réels et un préordre sur .

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.