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Order Complete
In mathematics, specifically in order theory and functional analysis, a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A (meaning contained in an interval, which is a set of the form [a, b] := \, for some a, b \in A), the supremum \sup S' and the infimum \inf S both exist and are elements of A. 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 case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices. Examples The Order dual (functional analysis), order dual of a vector lattice is an order complete vector lattice under its canonica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition Given a vector space ''X'' over the real numbers R and a preorder ≤ on the set ''X'', the pair is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of ''X'' and call ≤ a vector preorder on ''X'' if for all ''x'', ''y'', ''z'' in ''X'' and ''λ'' in R with the following two axioms are satisfied # implies # implies . If ≤ is a partial order compatible with the vector space structure of ''X'' then is called an ordered vector space and ≤ is called a vector partial order on ''X''. 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. Ordered vector spaces are ordered groups under their addition op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Lattice
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. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis. Definition Preliminaries If X is an ordered vector space (which by definition is a vector space over the reals) and if S is a subset of X then an element b \in X is an upper bound (resp. lower bound) of S if s \leq b (resp. s \geq b) for all s \in S. An element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Lattice
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order \,\leq\, 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. Definition If X is a vector lattice then by the vector lattice operations we mean the following maps: # the three maps X to itself defined by x \mapsto, x , , x \mapsto x^+, x \mapsto x^, and # the two maps from X \times X into X defined by (x, y) \mapsto \sup_ \ and(x, y) \mapsto \inf_ \. If X is a TVS over the reals and a vector lattice, then X is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous. If X 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Dual (functional Analysis)
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X is the set \operatorname\left(X^*\right) - \operatorname\left(X^*\right) where \operatorname\left(X^*\right) denotes the set of all positive linear functionals on X, where a linear function f on X is called positive if for all x \in X, x \geq 0 implies f(x) \geq 0. The order dual of X is denoted by X^+. Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces. Canonical ordering An element f of the order dual of X is called positive if x \geq 0 implies \operatorname f(x) \geq 0. The positive elements of the order dual form a cone that induces an ordering on X^+ called the canonical ordering. If X is an ordered vector space whose positive cone C is generating (that is, X = C - C) then the order dual with the canonical ordering is an ordered vector space. The order dual is the span ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable topologies on vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Disjoint
In mathematics, specifically in order theory and functional analysis, two elements ''x'' and ''y'' of a vector lattice ''X'' are lattice disjoint or simply disjoint if \inf \left\ = 0, in which case we write x \perp y, where the absolute value of ''x'' is defined to be , x, := \sup \left\. We say that two sets ''A'' and ''B'' are lattice disjoint or disjoint if ''a'' and ''b'' are disjoint for all ''a'' in ''A'' and all ''b'' in ''B'', in which case we write A \perp B. If ''A'' is the singleton set \ then we will write a \perp B in place of \ \perp B. For any set ''A'', we define the disjoint complement to be the set A^ := \left\. Characterizations Two elements ''x'' and ''y'' are disjoint if and only if \sup\ = , x , + , y , . If ''x'' and ''y'' are disjoint then , x + y , = , x , + , y , and \left(x + y \right)^ = x^ + y^, where for any element ''z'', z^ := \sup \left\ and z^ := \sup \left\. Properties Disjoint complements are always bands, but the converse is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
