|
Regularly Ordered
In mathematics, specifically in order theory and functional analysis, an ordered vector space X is said to be regularly ordered and its order is called regular if X is Archimedean ordered and the order dual of X distinguishes points in X. Being a regularly ordered vector space is an important property in the theory of topological vector lattices. Examples Every ordered locally convex space is regularly ordered. The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered. Properties If X is a regularly ordered vector lattice then the order topology on X is the finest topology on X making X into a locally convex 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 ba .... See also ... [...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 differe ... [...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, norm, topology, etc.) and the 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 spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the ... [...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 additio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Ordered
In mathematics, specifically in order theory, a binary relation \,\leq\, on a vector space X over the real or complex numbers is called Archimedean if for all x \in X, whenever there exists some y \in X such that n x \leq y for all positive integers n, then necessarily x \leq 0. An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space X is called almost Archimedean if for all x \in X, whenever there exists a y \in X such that -n^ y \leq x \leq n^ y for all positive integers n, thenx = 0. Characterizations A preordered vector space (X, \leq) with an order unit u is Archimedean preordered if and only if n x \leq u for all non-negative integers n implies x \leq 0. Properties Let X be an ordered vector space over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS. Order unit norm ... [...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] |
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 th ... [...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 elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Topology (functional Analysis)
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space (X, \leq) is the finest locally convex topological vector space (TVS) topology on X for which every order interval is bounded, where an order interval in X is a set of the form , b:= \left\ where a and b belong to X. 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 (X, \leq), rather than from some topology that X starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of (X, \leq). For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology. Definitions The family of all locally convex topologies on X for which every order interval is bounded is non-empty (sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
