Modes of convergence (annotated index)

The purpose of this article is to serve as an annotated

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An annotation is a note that is made while reading any form of text. This may be as simple as underlining or highlighting passages.Annotated bibliographies give descriptions about how each source is useful to an author in constructing a paper or argument...

index

Index (publishing)

An index is a list of words or phrases and associated pointers to where useful material relating to that heading can be found in a document...

of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence

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 of convergence in the settings where they are defined...

. 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.

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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 group

In mathematics, a topological abelian group, or TAG, is a topological group that is also an abelian group.That is, a TAG is both a group and a topological space, the group operations are continuous, and the group's binary operation is commutative....

(topological abelian groups), normed spaces

Normed vector space

In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

, Euclidean spaces, and the real/complex numbers. Also note that any metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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:

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

Convergence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

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

...in a uniform space (U)

Cauchy-convergence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

Implications:

- Convergence

Cauchy-convergence

- Cauchy-convergence and convergence of a subsequence together

convergence.

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

convergence.

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

A series of elements Σb_k in a TAG (G)

Convergence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

(of partial sum sequence)
Cauchy-convergence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

(of partial sum sequence)
Unconditional convergence

Implications:

- Unconditional convergence

convergence (by definition).

...in a normed space (N)

Absolute-convergence
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

(convergence of )

Implications:

- Absolute-convergence

Cauchy-convergence

absolute-convergence of some grouping¹.

- Therefore: N is Banach

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

(complete) if absolute-convergence

convergence.

- Absolute-convergence and convergence together

unconditional convergence.

- Unconditional convergence

absolute-convergence, even if N is Banach.

- If N is a Euclidean space, then unconditional convergence

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 {f_n} from a set (S) to a topological space (Y)

Pointwise convergence
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

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

Uniform convergence
Pointwise Cauchy-convergence
Uniform Cauchy-convergence

Implications are cases of earlier ones, except:

- Uniform convergence

both pointwise convergence and uniform Cauchy-convergence.

- Uniform Cauchy-convergence and pointwise convergence of a subsequence

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
Compact convergence
In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.-Definition:...

(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

both local uniform convergence and compact (uniform) convergence.

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

Local uniform convergence

compact (uniform) convergence.

- If

is locally compact, the converses to such tend to hold:

Local uniform convergence

compact (uniform) convergence.

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

Almost everywhere convergence

Almost uniform convergence

L^p convergence

Convergence in measure
Convergence in measure
Convergence in measure can refer to two distinct mathematical concepts which both generalizethe concept of convergence in probability.-Definitions:...

Convergence in distribution

Implications:

- Pointwise convergence

almost everywhere convergence.

- Uniform convergence

almost uniform convergence.

- Almost everywhere convergence

convergence in measure. (In a finite measure space)

- Almost uniform convergence

convergence in measure.

- L^p convergence

convergence in measure.

- Convergence in measure

convergence in distribution if μ is a probability measure and the functions are integrable.

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

Pointwise convergence
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

(of partial sum sequence)

Uniform convergence (of partial sum sequence)

Pointwise Cauchy-convergence (of partial sum sequence)

Uniform Cauchy-convergence (of partial sum sequence)

Unconditional pointwise convergence

Unconditional uniform convergence

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 in place of .

Pointwise absolute-convergence (pointwise convergence of )

Uniform absolute-convergence
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 :...

(uniform convergence of )

Normal convergence
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 :...

^{http://eom.springer.de/N/n067430.htm} (convergence of the series of uniform norms )

Implications are cases of earlier ones, except:

- Normal convergence uniform absolute-convergence

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

Local uniform convergence (of partial sum sequence)

Compact (uniform) convergence
Compact convergence
In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.-Definition:...

(of partial sum sequence)