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, 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   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 Σbk in a TAG (G)Edit

Implications:

  -   Unconditional convergence   convergence (by definition).

...in a normed space (N)Edit

Implications:

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

  -   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)Edit

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

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)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   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)Edit

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.

  -   Lp 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 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   in place of  .

  • Pointwise absolute-convergence (pointwise convergence of  )
  • Uniform absolute-convergence (uniform convergence of  )
  • Normal convergence (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)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   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 alsoEdit