Modes of convergence (annotated index)

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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, 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 superheadings.

The following is a list of modes of convergence for:

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

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

...in a uniform space (U)[edit]

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)[edit]

Implications:

  -   Unconditional convergence \Rightarrow convergence (by definition).

...in a normed space (N)[edit]

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)[edit]

...from a set (S) to a uniform space (U)[edit]

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)[edit]

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:

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)[edit]

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)[edit]

Implications are all cases of earlier ones.

...from a set (S) to a normed space (N)[edit]

Generally, replacing "convergence" by "absolute-convergence" means one is referring to convergence of the series of nonnegative functions \Sigma|g_k| in place of \Sigma g_k.

Implications are cases of earlier ones, except:

  -   Normal convergence \Rightarrow uniform absolute-convergence

...from a topological space (X) to a TAG (G)[edit]

Implications are all cases of earlier ones.

...from a topological space (X) to a normed space (N)[edit]

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[edit]