|
Abstract L-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 extre ... [...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] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Probability Space
Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object that bears a defined relationship to a unit of measure used for calibration of measuring devices * Standard (timber unit), an obsolete measure of timber used in trade * Breed standard (also called bench standard), in animal fancy and animal husbandry * BioCompute Standard, a standard for next generation sequencing * ''De facto'' standard, product or system with market dominance * Gold standard, a monetary system based on gold; also used metaphorically for the best of several options, against which the others are measured * Internet Standard, a specification ratified as an open standard by the Internet Engineering Task Force * Learning standards, standards applied to education content * Standard displacement, a naval term describing the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 X 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] |
