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Modes Of Convergence
In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index) Note that each of the following objects is a special case of the types preceding it: Set (mathematics), sets, topological spaces, uniform spaces, topological abelian group, TAGs (topological abelian groups), Normed vector space, normed spaces, Euclidean spaces, and the real/complex numbers. Also, note that any metric space is a uniform space. Elements of a topological space Convergence can be defined in terms of Limit of a sequence, sequences in first-countable spaces. Net (mathematics), Nets are a generalization of sequences that are useful in spaces which are not first countable. Filter (set theory), Filters further generalize the concept of convergence. In metric spaces, one can define Cauchy s ... [...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] |
Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Convergence
In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. History The concept of normal convergence was first introduced by René Baire in 1908 in his book ''Leçons sur les théories générales de l'analyse''. Definition Given a set ''S'' and functions f_n : S \to \mathbb (or to any normed vector space), the series :\sum_^\infty f_n(x) is called normally convergent if the series of uniform norms of the terms of the series converges, i.e., :\sum_^\infty \, f_n\, := \sum_^\infty \sup_ , f_n(x), < \infty. Distinctions Normal convergence implies, but should not be confused with, , i.e. uniform convergence of the series of nonn ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Linear Space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbourhoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Convergence
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Definition Let (X, \mathcal) be a topological space and (Y,d_) be a metric space. A sequence of functions :f_ : X \to Y, n \in \mathbb, is said to converge compactly as n \to \infty to some function f : X \to Y if, for every compact set K \subseteq X, :f_, _ \to f, _ uniformly on K as n \to \infty. This means that for all compact K \subseteq X, :\lim_ \sup_ d_ \left( f_ (x), f(x) \right) = 0. Examples * If X = (0, 1) \subseteq \mathbb and Y = \mathbb with their usual topologies, with f_ (x) := x^, then f_ converges compactly to the constant function with value 0, but not uniformly. * If X=(0,1], Y=\R and f_n(x)=x^n, then f_n converges pointwise convergence, pointwise to the function that is zero on (0,1) and one at 1, but the sequence does not converge compactly. * A very po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Uniform Convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily small positive number \epsilon, a number N can be found such that each of the functions f_N, f_,f_,\ldots differs from f by no more than \epsilon ''at every point'' x ''in'' E. Described in an informal way, if f_n converges to f uniformly, then the rate at which f_n(x) approaches f(x) is "uniform" throughout its domain in the following sense: in order to guarantee that f_n(x) falls within a certain distance \epsilon of f(x), we do not need to know the value of x\in E in question — there can be found a single value of N=N(\epsilon) ''independent of x'', such that choosing n\geq N will ensure that f_n(x) is within \epsilon of f(x) ''for all x\in E''. In contrast, pointwise convergence of f_n to f merely guarantees that for any x\in E given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Cauchy Sequence
In mathematics, a sequence of functions \ from a set ''S'' to a metric space ''M'' is said to be uniformly Cauchy if: * For all \varepsilon > 0, there exists N>0 such that for all x\in S: d(f_(x), f_(x)) N. Another way of saying this is that d_u (f_, f_) \to 0 as m, n \to \infty, where the uniform distance d_u between two functions is defined by :d_ (f, g) := \sup_ d (f(x), g(x)). Convergence criteria A sequence of functions from ''S'' to ''M'' is pointwise Cauchy if, for each ''x'' ∈ ''S'', the sequence is a Cauchy sequence in ''M''. This is a weaker condition than being uniformly Cauchy. In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space ''M'' is complete, then any pointwise Cauchy sequence converges pointwise to a function from ''S'' to ''M''. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function. The uniform Cauchy prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily small positive number \epsilon, a number N can be found such that each of the functions f_N, f_,f_,\ldots differs from f by no more than \epsilon ''at every point'' x ''in'' E. Described in an informal way, if f_n converges to f uniformly, then the rate at which f_n(x) approaches f(x) is "uniform" throughout its domain in the following sense: in order to guarantee that f_n(x) falls within a certain distance \epsilon of f(x), we do not need to know the value of x\in E in question — there can be found a single value of N=N(\epsilon) ''independent of x'', such that choosing n\geq N will ensure that f_n(x) is within \epsilon of f(x) ''for all x\in E''. In contrast, pointwise convergence of f_n to f merely guarantees that for any x\in E given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise Convergence
In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and Y is a topological space, such as the Real number, real or complex numbers or a metric space, for example. A Net (mathematics), net or sequence of Function (mathematics), functions \left(f_n\right) all having the same domain X and codomain Y is said to converge pointwise to a given function f : X \to Y often written as \lim_ f_n = f\ \mbox if (and only if) \lim_ f_n(x) = f(x) \text x \text f. The function f is said to be the pointwise limit function of the \left(f_n\right). Sometimes, authors use the term bounded pointwise convergence when there is a constant C such that \forall n,x,\;, f_n(x), Properties This concept is often contra ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of A Function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both subsets of \R, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the codomain, and the set of values attained by the function (which is a subset of ) is called its range or image. Any function can be restricted to a subset of its domain. The restriction of f \colon X \to Y to A, where A\subseteq X, is written as \left. f \_A \colon A \to Y. Natural domain If a real function is giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
