Concept
Riesz space
Concepts associés (6)
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 .
Order dual (functional analysis)
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all implies The order dual of is denoted by . Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.
Order complete
In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (meaning contained in an interval, which is a set of the form for some ), the supremum ' and the infimum both exist and are elements of An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which ca
Archimedean ordered vector space
In 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.
Order bound dual
In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space is the set of all linear functionals on that map order intervals, which are sets of the form to bounded sets. The order bound dual of is denoted by This space plays an important role in the theory of ordered topological vector spaces. An element of the order bound dual of is called positive if implies The positive elements of the order bound dual form a cone that induces an ordering on called the .
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.