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Banach Limit
In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\infty, and complex numbers \alpha: # \phi(\alpha x+y) = \alpha\phi(x) + \phi(y) (linearity); # if x_n\geq 0 for all n \in \mathbb, then \phi(x) \geq 0 (positivity); # \phi(x) = \phi(Sx), where S is the shift operator defined by (Sx)_n=x_ (shift-invariance); # if x is a convergent sequence, then \phi(x) = \lim x . Hence, \phi is an extension of the continuous functional \lim: c \to \mathbb C where c \subset\ell^\infty is the complex vector space of all sequences which converge to a (usual) limit in \mathbb C. In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Čech Compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone–Čech compactification ''βX'' of a topological space ''X'' is the largest, most general compact Hausdorff space "generated" by ''X'', in the sense that any continuous map from ''X'' to a compact Hausdorff space factors through ''βX'' (in a unique way). If ''X'' is a Tychonoff space then the map from ''X'' to its image in ''βX'' is a homeomorphism, so ''X'' can be thought of as a (dense) subspace of ''βX''; every other compact Hausdorff space that densely contains ''X'' is a quotient of ''βX''. For general topological spaces ''X'', the map from ''X'' to ''βX'' need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Formal definition Let X be a locally compact Hausdorff space, and let \mathfrak(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure \mu defined on the σ-algebra of Borel sets. A few authors require in addition that \mu is locally finite, meaning that \mu(C) 0 and μ(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the Hausdorff dimension dimHaus(''X'') ≥ ''s''. A partial converse is provided by the Frostman lemma: Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Additive
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstrac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures". Given a measurable space (X, \Sigma) (that is, a set X with a σ-algebra \Sigma on it), an extended signed measure is a set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ... \mu : \Sigma \to \R \cup \ such that \mu(\varnothing) = 0 and \mu is sigma additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ba Space
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X). If Σ is a sigma-algebra, then the space ca(\Sigma) is defined as the subset of ba(\Sigma) consisting of countably additive measures. The notation ''ba'' is a mnemonic for ''bounded additive'' and ''ca'' is short for ''countably additive''. If ''X'' is a topological space, and Σ is the sigma-algebra of Borel sets in ''X'', then rca(X) is the subspace of ca(\Sigma) consisting of all regular Borel measures on ''X''. Properties All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus ca(\Sigma) is a closed subset of ba(\Sigma), and rca(X) is a closed set of ca(\Sigma) for Σ the algebra of Borel sets on ''X''. The space of simple functions on \Sigma is dense in ba(\Sigma). T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
