Archimedean ordered vector spaceIn mathematics, specifically in order theory, a binary relation on a vector space over the real or complex numbers is called Archimedean if for all whenever there exists some such that for all positive integers then necessarily An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space is called almost Archimedean if for all whenever there exists a such that for all positive integers then A preordered vector space with an order unit is Archimedean preordered if and only if for all non-negative integers implies Let be an ordered vector space over the reals that is finite-dimensional.
Solid setIn mathematics, specifically in order theory and functional analysis, a subset of a vector lattice is said to be solid and is called an ideal if for all and if then An ordered vector space whose order is Archimedean is said to be Archimedean ordered. If then the ideal generated by is the smallest ideal in containing An ideal generated by a singleton set is called a principal ideal in The intersection of an arbitrary collection of ideals in is again an ideal and furthermore, is clearly an ideal of itself; thus every subset of is contained in a unique smallest ideal.
Espace vectoriel ordonnéEn 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 .
Order topology (functional analysis)In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space is the finest locally convex topological vector space (TVS) topology on for which every order interval is bounded, where an order interval in is a set of the form where and belong to The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of rathe
Riesz spaceIn 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.
