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.

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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, topological abelian groups (TAG), normed vector 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 super headings.

The following is a list of modes of convergence for:

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

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

...in a uniform space (U)

• Cauchy-convergence

Implications:

  -   Convergence ${\displaystyle \Rightarrow }$  Cauchy-convergence

  -   Cauchy-convergence and convergence of a subsequence together ${\displaystyle \Rightarrow }$  convergence.

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

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

A series of elements Σb_k in a TAG (G)

• Convergence (of partial sum sequence)
• Cauchy-convergence (of partial sum sequence)
• Unconditional convergence

Implications:

  -   Unconditional convergence ${\displaystyle \Rightarrow }$  convergence (by definition).

...in a normed space (N)

• Absolute-convergence (convergence of ${\displaystyle \sum |b_{k}|}$ )

Implications:

  -   Absolute-convergence ${\displaystyle \Rightarrow }$  Cauchy-convergence ${\displaystyle \Rightarrow }$  absolute-convergence of some grouping¹.

  -   Therefore: N is Banach (complete) if absolute-convergence ${\displaystyle \Rightarrow }$  convergence.

  -   Absolute-convergence and convergence together ${\displaystyle \Rightarrow }$  unconditional convergence.

  -   Unconditional convergence ${\displaystyle \not \Rightarrow }$  absolute-convergence, even if N is Banach.

  -   If N is a Euclidean space, then unconditional convergence ${\displaystyle \equiv }$  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 {f_n} from a set (S) to a topological space (Y)

• Pointwise convergence

...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 ${\displaystyle \Rightarrow }$  both pointwise convergence and uniform Cauchy-convergence.

  -   Uniform Cauchy-convergence and pointwise convergence of a subsequence ${\displaystyle \Rightarrow }$  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 ${\displaystyle \Rightarrow }$  both local uniform convergence and compact (uniform) convergence.

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

      Local uniform convergence ${\displaystyle \Rightarrow }$  compact (uniform) convergence.

  -   If ${\displaystyle X}$  is locally compact, the converses to such tend to hold:

      Local uniform convergence ${\displaystyle \equiv }$  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 distribution

Implications:

  -   Pointwise convergence ${\displaystyle \Rightarrow }$  almost everywhere convergence.

  -   Uniform convergence ${\displaystyle \Rightarrow }$  almost uniform convergence.

  -   Almost everywhere convergence ${\displaystyle \Rightarrow }$  convergence in measure. (In a finite measure space)

  -   Almost uniform convergence ${\displaystyle \Rightarrow }$  convergence in measure.

  -   L^p convergence ${\displaystyle \Rightarrow }$  convergence in measure.

  -   Convergence in measure ${\displaystyle \Rightarrow }$  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 (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 ${\displaystyle \Sigma |g_{k}|}$  in place of ${\displaystyle \Sigma g_{k}}$ .

• Pointwise absolute-convergence (pointwise convergence of ${\displaystyle \Sigma |g_{k}|}$ )
• Uniform absolute-convergence (uniform convergence of ${\displaystyle \Sigma |g_{k}|}$ )
• Normal convergence (convergence of the series of uniform norms ${\displaystyle \Sigma ||g_{k}||_{u}}$ )

Implications are cases of earlier ones, except:

  -   Normal convergence ${\displaystyle \Rightarrow }$  uniform absolute-convergence

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

• Local uniform convergence (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 ${\displaystyle \Rightarrow }$  both local uniform absolute-convergence and compact (uniform) absolute-convergence.

      Normal convergence ${\displaystyle \Rightarrow }$  both local normal convergence and compact normal convergence.

  -   Local normal convergence ${\displaystyle \Rightarrow }$  local uniform absolute-convergence.

      Compact normal convergence ${\displaystyle \Rightarrow }$  compact (uniform) absolute-convergence.

  -   Local uniform absolute-convergence ${\displaystyle \Rightarrow }$  compact (uniform) absolute-convergence.

      Local normal convergence ${\displaystyle \Rightarrow }$  compact normal convergence

  -   If X is locally compact:

      Local uniform absolute-convergence ${\displaystyle \equiv }$  compact (uniform) absolute-convergence.

      Local normal convergence ${\displaystyle \equiv }$  compact normal convergence

See also

• Limit of a sequence
• Convergence of measures
• Convergence in measure
• Convergence of random variables:
  • in distribution
  • in probability
  • almost sure
  • sure
  • in mean

References