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Subsequential Limit
In mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In first-countable spaces, the two concepts coincide. In a topological space, if every subsequence has a subsequential limit to the same point, then the original sequence also converges to that limit. This need not hold in more generalized notions of convergence, such as the space of almost everywhere convergence. The supremum of the set of all subsequential limits of some sequence is called the limit superior, or limsup. Similarly, the infimum of such a set is called the limit inferior, or liminf. See limit superior and limit inferior. If (X, d) is a metric space and there is a Cauchy sequence 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 represe ... [...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] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a preorder. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as \langle B,C,D \rangle, from \langle A,B,C,D,E,F \rangle, is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "''a''", "''ap''", "''al''", "''ae''", "''app''", "''apl''", "''ape''", "''ale''", "''appl''", "''appe''", "''aple''", "''apple''", "''p''", "''pp''", "''pl''", "''pe''", "''ppl''", "''ppe''", "''ple' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence (x_n)_ in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_n \in V. This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-countable Space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence N_1, N_2, \ldots of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with N_i contained in N. Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods. Examples and counterexamples The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers form a countable local base at x. An example of a space which is not first-countable is the cofinite topology on an uncountable set ( ... [...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] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Superior And Limit Inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence x_n is denoted by \liminf_x_n\quad\text\quad \varliminf_x_n. The limit superior of a sequence x_n is denoted by \lims ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Sequence
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 ..., a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose Element (mathematics), elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: a_n=\sqrt n, the consecutive terms become arbitrarily close to each other: a_-a_n = \sqrt-\sqrt = \frac d. (Actually, any m > \left(\sqrt + d\right)^2 suffices.) As a result, despite how far one goes, the remaining terms of the sequence never get c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limits (mathematics)
Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 song by Paenda; see Austria in the Eurovision Song Contest 2019 * ''Limits'' (collection), a collection of short stories and essays by Larry Niven * The Limit, a Dutch band *The Limit, an episode from ''The Amazing World of Gumball'' Mathematics * Limit (mathematics), the value that a function or sequence "approaches" as the input or index approaches some value ** Limit of a function ***(ε,_δ)-definition of limit, formal definition of the mathematical notion of limit ** Limit of a sequence ** One-sided limit, either of the two limits of a function as a specified point is approached from below or from above * Limit of a net * Limit point, in topological spaces * Limit (category theory) ** Direct limit ** Inverse limit Other uses * Limi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
