The purpose of this article is to serve as an
annotated
An annotation is extra information associated with a particular point in a document or other piece of information. It can be a note that includes a comment or explanation. Annotations are sometimes presented in the margin of book pages. For ann ...
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
of various modes of convergence and their logical relationships. For an expository article, see
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, se ...
. 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:
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s,
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s,
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s,
topological abelian group
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
s (TAG),
normed vector 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" i ...
s,
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s, and the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
/
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers. Also note that any
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 settin ...
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 in a topological space (''Y'')
*
Convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
, or "topological convergence" for emphasis (i.e. the existence of a limit).
...in a uniform space (''U'')
*
Cauchy-convergence
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 Σ''bk'' in a TAG (''G'')
*
Convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
(of partial sum sequence)
*
Cauchy-convergence (of partial sum sequence)
*
Unconditional convergence In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not a ...
Implications:
- Unconditional convergence
convergence (by definition).
...in a normed space (''N'')
*
Absolute-convergence (convergence of
)
Implications:
- Absolute-convergence
Cauchy-convergence
absolute-convergence of some grouping
1.
- Therefore: ''N'' is
Banach (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 {''fn''} from a set (''S'') to a topological space (''Y'')
*
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can 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 ...
...from a set (''S'') to a uniform space (''U'')
*
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 s ...
* 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
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 s ...
(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
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
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can 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 ...
*
Almost uniform convergence
*
Lp convergence
*
Convergence in measure
Convergence in measure is either of two distinct mathematical concepts both of which generalize
the concept of convergence in probability.
Definitions
Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be measurable functions on a measure space (X, \ ...
*
Convergence in distribution
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications t ...
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 Σ''gk'' from a set (''S'') to a TAG (''G'')
*
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can 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 ...
(of partial sum sequence)
*
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 s ...
(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 In mathematics, uniform absolute-convergence is a type of Convergent series, convergence for series (mathematics), series of function (mathematics), functions. Like absolute convergence, absolute-convergence, it has the useful property that it is p ...
(uniform convergence of
)
*
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 ...
(convergence of the series of
uniform norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
s
)
Implications are cases of earlier ones, except:
- Normal convergence
uniform absolute-convergence
...from a topological space (''X'') to a TAG (''G'')
*
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 s ...
(of partial sum sequence)
*
Compact (uniform) convergence (of partial sum sequence)
Implications are all cases of earlier ones.
...from a topological space (''X'') to a normed space (''N'')
*
Local uniform absolute-convergence
*
Compact (uniform) absolute-convergence
*
Local normal convergence
*
Compact normal convergence
Implications (mostly cases of earlier ones):
- Uniform absolute-convergence
both local uniform absolute-convergence and compact (uniform) absolute-convergence.
Normal convergence
both local normal convergence and compact normal convergence.
- Local normal convergence
local uniform absolute-convergence.
Compact normal convergence
compact (uniform) absolute-convergence.
- Local uniform absolute-convergence
compact (uniform) absolute-convergence.
Local normal convergence
compact normal convergence
- If ''X'' is locally compact:
Local uniform absolute-convergence
compact (uniform) absolute-convergence.
Local normal convergence
compact normal convergence
See also
*
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 limi ...
*
Convergence of measures
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...
*
Convergence in measure
Convergence in measure is either of two distinct mathematical concepts both of which generalize
the concept of convergence in probability.
Definitions
Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be measurable functions on a measure space (X, \ ...
*
Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
:
**
in distribution
**
in probability
**
almost sure
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
**
sure
**
in mean
Convergence (mathematics)