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AM-space
In mathematics, specifically in order theory and functional analysis, an abstract ''m''-space or an AM-space is a Banach lattice (X, \, \cdot \, ) whose norm satisfies \left\, \sup \ \right\, = \sup \left\ for all ''x'' and ''y'' in the positive cone of ''X''. We say that an AM-space ''X'' is an AM-space with unit if in addition there exists some in ''X'' such that the interval is equal to the unit ball of ''X''; such an element ''u'' is unique and an order unit of ''X''. Examples The strong dual of an AL-space is an AM-space with unit. If ''X'' is an Archimedean ordered vector lattice, ''u'' is an order unit of ''X'', and ''p''''u'' is the Minkowski functional of , -u:= \, then the complete of the semi-normed space (''X'', ''p''''u'') is an AM-space with unit ''u''. Properties Every AM-space is isomorphic (as a Banach lattice) with some closed vector sublattice of some suitable C_\left( X \right). The strong dual of an AM-space with unit is an AL-space. If ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AL-space
In mathematics, specifically in order theory and functional analysis, an abstract ''L''-space, an AL-space, or an abstract Lebesgue space is a Banach lattice (X, \, \cdot \, ) whose norm is additive on the positive cone of ''X''. In probability theory, it means the standard probability space. Examples The strong dual of an AM-space with unit is an AL-space. Properties The reason for the name abstract ''L''-space is because every AL-space is isomorphic (as a Banach lattice) with some subspace of L^1(\mu). Every AL-space ''X'' is an order complete vector lattice of minimal type; however, the order dual of ''X'', denoted by ''X''+, is ''not'' of minimal type unless ''X'' is finite-dimensional. Each order interval in an AL-space is weakly compact. The strong dual of an AL-space is an AM-space with unit. The continuous dual space X^ (which is equal to ''X''+) of an AL-space ''X'' is a Banach lattice that can be identified with C_ ( K ), where ''K'' is a compact extremall ... [...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 in general (although one usually is also interested in the actual difference of two numbers, which is ... [...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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. 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, for example, continuous function, continuous or unitary operator, unitary 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 v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Lattice
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, \leq, such that for all , the implication \Rightarrow holds, where the absolute value is defined as , x, = x \vee -x := \sup\\text Examples and constructions Banach lattices are extremely common in functional analysis, and "every known example n 1948of a Banach space asalso a vector lattice." In particular: * , together with its absolute value as a norm, is a Banach lattice. * Let be a topological space, a Banach lattice and the space of continuous bounded functions from to with norm \, f\, _ = \sup_ \, f(x)\, _Y\text Then is a Banach lattice under the pointwise partial order: \Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text Examples of non-lattice Banach spaces are now known; James' space is one such.Kania, Tomasz (12 April 2017).Answerto "Banach space that is not a Banach lattice" (accessed 13 August 2022). ... [...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 a 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 topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Order
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, as 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 ''comparable'', in the sens ... [...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 eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then the or of K is defined to be the function p_K : X \to , \infty valued in the extended real numbers, defined by p_K(x) := \inf \ \quad \text x \in X, where the infimum of the empty set is defined to be positive infinity \,\infty\, (which is a real number so that p_K(x) would then be real-valued). The set K is often assumed/picked to have properties, such as being an absorbing disk in X, that guarantee that p_K will be a real-valued seminorm on X. In fact, every seminorm p on X is equal to the Minkowski functional (that is, p = p_K) of any subset K of X satisfying \ \subseteq K \subseteq \ (where all three of these sets are necessarily absorbing in X and the first and last are also disks). Thus every seminorm (which is a d ... [...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 eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
