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Riesz Space
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] |
Freudenthal Spectral Theorem
In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions. Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known Radon–Nikodym theorem, the validity of the Poisson formula and the spectral theorem from the theory of normal operators can all be shown to follow as special cases of the Freudenthal spectral theorem. Statement Let ''e'' be any positive element in a Riesz space ''E''. A positive element of ''p'' in ''E'' is called a component of ''e'' if p\wedge(e-p)=0. If p_1,p_2,\ldots,p_n are pairwise disjoint components of ''e'', any real linear combination of p_1,p_2,\ldots,p_n is called an ''e''-simple function. The Freudenthal spectral theorem states: Let ''E'' be any Riesz space with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Join Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa. Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order. A lattice is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Order
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] |
Archimedean Property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers ''x'' and ''y'', there is an integer ''n'' such that ''nx'' > ''y''. It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ ''On the Sphere and Cylinder''. The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure in which any two non-zero elements are ''comparabl ... [...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 [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] |
Algebraic Dual
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (wh ... [...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] |
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. * Indexing int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing Set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a Set (mathematics), set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are Radial set, radial or absorbent set. Every Neighbourhood (mathematics), neighborhood of the origin in every topological vector space is an absorbing subset. Definition Suppose that X is a vector space over the Field (mathematics), field \mathbb of real numbers \R or complex numbers \Complex. Notation Products of scalars and vectors For any -\infty \leq r \leq R \leq \infty, vector x, and subset A \subseteq X, let B_r = \ \quad \text \quad B_ = \ denote the ''open ball'' (respectively, the ''closed ball'') of radius r in \mathbb centered at 0, and let (r, R) x = \ \quad \text \quad (r, R) A = \. Similarly, if K \subseteq \mathbb and k is a scalar then let K A = \, K x = \, k A = \, and \mathbb x = \ = \operatorname \. Balanced co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Unit
An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first example below) the order unit generalizes the unit element in the reals. According to H. H. Schaefer, "most of the ordered vector spaces occurring in analysis do not have order units." Definition For the ordering cone K \subseteq X in the vector space X, the element e \in K is an order unit (more precisely an K-order unit) if for every x \in X there exists a \lambda_x > 0 such that \lambda_x e - x \in K (that is, x \leq_K \lambda_x e). Equivalent definition The order units of an ordering cone K \subseteq X are those elements in the algebraic interior of K; that is, given by \operatorname(K). Examples Let X = \R be the real numbers and K = \R_+ = \, then the unit element 1 is an . Let X = \R^n and K = \R^n_+ = \left\, then the unit element \vec = (1, \ldots, 1) is an . Each interior point of the positive cone of an ordered topolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \leq 1. The balanced hull or balanced envelope of a set S is the smallest balanced set containing S. The balanced core of a subset S is the largest balanced set contained in S. Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Definition Let X be a vector space over the field \mathbb of real or complex numbers. Notation If S is a set, a is a scalar, and B \subseteq \mathbb then let a S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
