|
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] |
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 differenc ... [...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 I ... [...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] |
Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples * The constant zero function, mapping every vector to zero, is trivially a linear functional. * Index ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Bound Dual
In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space X is the set of all linear functionals on X that map order intervals, which are sets of the form , b:= \, to bounded sets. The order bound dual of X is denoted by X^. This space plays an important role in the theory of ordered topological vector spaces. Canonical ordering An element g of the order bound dual of X is called positive if x \geq 0 implies \operatorname(f(x)) \geq 0. The positive elements of the order bound dual form a cone that induces an ordering on X^ called the . If X is an ordered vector space whose positive cone C is generating (meaning X = C - C) then the order bound dual with the canonical ordering is an ordered vector space. Properties The order bound dual of an ordered vector spaces contains its order dual. If the positive cone of an ordered vector space X is generating and if for all positive x and x we have , x+ , y= , x + y then th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Topological Vector Space
In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) ''X'' that has a partial order ≤ making it into an ordered vector space whose positive cone C := \left\ is a closed subset of ''X''. Ordered TVS have important applications in spectral theory. Normal cone If ''C'' is a cone in a TVS ''X'' then ''C'' is normal if \mathcal = \left \mathcal \right, where \mathcal is the neighborhood filter at the origin, \left \mathcal \right = \left\, and := \left(U + C\right) \cap \left(U - C\right) is the ''C''-saturated hull of a subset ''U'' of ''X''. If ''C'' is a cone in a TVS ''X'' (over the real or complex numbers), then the following are equivalent: # ''C'' is a normal cone. # For every filter \mathcal in ''X'', if \lim \mathcal = 0 then \lim \left \mathcal \right = 0. # There exists a neighborhood base \mathcal in ''X'' such that B \in \mathcal implies \left B \cap ... [...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 ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Set
In mathematics, specifically in order theory and functional analysis, a subset S of a vector lattice is said to be solid and is called an ideal if for all s \in S and x \in X, if , x, \leq , s, then x \in S. An 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 pr ... whose order is Archimedean is said to be '' Archimedean ordered''. If S\subseteq X then the ideal generated by S is the smallest ideal in X containing S. An ideal generated by a singleton set is called a principal ideal in X. Examples The intersection of an arbitrary collection of ideals in X is again an ideal and furthermore, X is clearly an ideal of itself; thus every subset of X is contained in a unique smallest ideal. In a locally convex vector lattice X, the polar of every solid neighborho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sublattice
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Convergent
In mathematics, specifically in order theory and functional analysis, a filter \mathcal in an order complete vector lattice X is order convergent if it contains an order bounded subset (that is, is contained in an interval of the form , b:= \) and if \mathcal, \sup \left\ = \inf \left\, where \operatorname(X) is the set of all order bounded subsets of ''X'', in which case this common value is called the order limit of \mathcal in X. Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology. Definition A net \left(x_\right)_ in a vector lattice X is said to decrease to x_0 \in X if \alpha \leq \beta implies x_ \leq x_ and x_0 = inf \left\ in X. A net \left(x_\right)_ in a vector lattice X is said to order-converge to x_0 \in X if there is a net \left(y_\right)_ in X that decreases to 0 and satisfies \left, x_ - x_0\ \leq y_ for all \alpha \in A. Order continuity A linear map T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 , 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 of a vector lattice is an order complete vector lattice under its canonical ordering. If X is a locally co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Order Unit
In mathematics, specifically in order theory and functional analysis, an element x of a 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 ''Su ... X is called a weak order unit in X if x \geq 0 and also for all y \in X, \inf \ = 0 \text y = 0. Examples * If X is a separable Fréchet topological vector lattice then the set of weak order units is dense in the positive cone of X. See also * * Citations References * * {{mathematics-stub Functional analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
