Modes of convergence (annotated index)


Modes of convergence (annotated index)

The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick reference, and indepth descriptions and discussions are reserved for their respective articles.


Guide to this index. To avoid excessive verbiage, note that each of the following types of objects is a special case of types preceding it: sets, topological spaces, uniform spaces, TAGs (topological abelian groups), normed spaces, Euclidean spaces, and the real/complex numbers. Also note that any metric space is a uniform space. Finally, subheadings will always indicate special cases of their superheadings.

The following is a list of modes of convergence for:

Contents

A sequence of elements {an} in a topological space (Y)

  • Convergence , or "topological convergence" for emphasis (i.e. the existence of a limit).

...in a uniform space (U)

Implications:

  -   Convergence \Rightarrow Cauchy-convergence

  -   Cauchy-convergence and convergence of a subsequence together \Rightarrow convergence.

  -   U is called "complete" if Cauchy-convergence (for nets) \Rightarrow convergence.

Note: A sequence exhibiting Cauchy-convergence is called a cauchy sequence to emphasize that it may not be convergent.

A series of elements Σbk in a TAG (G)

Implications:

  -   Unconditional convergence \Rightarrow convergence (by definition).

...in a normed space (N)

Implications:

  -   Absolute-convergence \Rightarrow Cauchy-convergence \Rightarrow absolute-convergence of some grouping1.

  -   Therefore: N is Banach (complete) if absolute-convergence \Rightarrow convergence.

  -   Absolute-convergence and convergence together \Rightarrow unconditional convergence.

  -   Unconditional convergence \not\Rightarrow absolute-convergence, even if N is Banach.

  -   If N is a Euclidean space, then unconditional convergence \equiv absolute-convergence.

1 Note: "grouping" refers to a series obtained by grouping (but not reordering) terms of the original series. A grouping of a series thus corresponds to a subsequence of its partial sums.

A sequence of functions {fn} from a set (S) to a topological space (Y)

...from a set (S) to a uniform space (U)

Implications are cases of earlier ones, except:

  -   Uniform convergence \Rightarrow both pointwise convergence and uniform Cauchy-convergence.

  -   Uniform Cauchy-convergence and pointwise convergence of a subsequence \Rightarrow uniform convergence.

...from a topological space (X) to a uniform space (U)

For many "global" modes of convergence, there are corresponding notions of a) "local" and b) "compact" convergence, which are given by requiring convergence to occur a) on some neighborhood of each point, or b) on all compact subsets of X. Examples:

  • Local uniform convergence (i.e. uniform convergence on a neighborhood of each point)
  • Compact (uniform) convergence (i.e. uniform convergence on all compact subsets)
  • further instances of this pattern below.

Implications:

  -   "Global" modes of convergence imply the corresponding "local" and "compact" modes of convergence. E.g.:

      Uniform convergence \Rightarrow both local uniform convergence and compact (uniform) convergence.

  -   "Local" modes of convergence tend to imply "compact" modes of convergence. E.g.,

      Local uniform convergence \Rightarrow compact (uniform) convergence.

  -   If X is locally compact, the converses to such tend to hold:

      Local uniform convergence \equiv compact (uniform) convergence.

...from a measure space (S,μ) to the complex numbers (C)

Implications:

  -   Pointwise convergence \Rightarrow almost everywhere convergence.

  -   Uniform convergence \Rightarrow almost uniform convergence.

  -   Almost everywhere convergence \Rightarrow convergence in measure. (In a finite measure space)

  -   Almost uniform convergence \Rightarrow convergence in measure.

  -   Lp convergence \Rightarrow convergence in measure.

  -   Convergence in measure \Rightarrow convergence in distribution if μ is a probability measure and the functions are integrable.

A series of functions Σgk from a set (S) to a TAG (G)

Implications are all cases of earlier ones.

...from a set (S) to a normed space (N)

Generally, replacing "convergence" by "absolute-convergence" means one is referring to convergence of the series of nonnegative functions Σ | gk | in place of Σgk.

Implications are cases of earlier ones, except:

  -   Normal convergence \Rightarrow uniform absolute-convergence

...from a topological space (X) to a TAG (G)

Implications are all cases of earlier ones.

...from a topological space (X) to a normed space (N)

Implications (mostly cases of earlier ones):

  -   Uniform absolute-convergence \Rightarrow both local uniform absolute-convergence and compact (uniform) absolute-convergence.

      Normal convergence \Rightarrow both local normal convergence and compact normal convergence.

  -   Local normal convergence \Rightarrow local uniform absolute-convergence.

      Compact normal convergence \Rightarrow compact (uniform) absolute-convergence.

  -   Local uniform absolute-convergence \Rightarrow compact (uniform) absolute-convergence.

      Local normal convergence \Rightarrow compact normal convergence

  -   If X is locally compact:

      Local uniform absolute-convergence \equiv compact (uniform) absolute-convergence.

      Local normal convergence \equiv compact normal convergence

See also


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • 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,… …   Wikipedia

  • Uniform convergence — In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does… …   Wikipedia

  • Absolute convergence — In mathematics, a series (or sometimes also an integral) of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex series is said to converge… …   Wikipedia

  • 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. Contents 1 History 2 Definition 3 Distinctions …   Wikipedia

  • Pointwise convergence — In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.[1][2] Contents 1 Definition 2 Properties …   Wikipedia

  • Uniform absolute-convergence — In mathematics, uniform absolute 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. Motivation A convergent series of… …   Wikipedia

  • Compact convergence — In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact open topology. Contents 1 Definition 2 Examples 3 Properties …   Wikipedia

  • Unconditional convergence — In mathematical analysis, a series sum {n=1}^infty x n in a Banach space X is unconditionally convergent if for every permutation sigma: mathbb N o mathbb N the series sum {n=1}^inftyx {sigma(n)} converges.This notion is often defined in an… …   Wikipedia

  • List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

  • Cauchy sequence — In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from… …   Wikipedia


We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.