 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, TAGs (topological abelian groups), normed 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:
Contents
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)
Implications:
 Convergence Cauchyconvergence
 Cauchyconvergence and convergence of a subsequence together convergence.
 U is called "complete" if Cauchyconvergence (for nets) convergence.
Note: A sequence exhibiting Cauchyconvergence 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)
 Cauchyconvergence (of partial sum sequence)
 Unconditional convergence
Implications:
 Unconditional convergence convergence (by definition).
...in a normed space (N)
 Absoluteconvergence (convergence of )
Implications:
 Absoluteconvergence Cauchyconvergence absoluteconvergence of some grouping^{1}.
 Therefore: N is Banach (complete) if absoluteconvergence convergence.
 Absoluteconvergence and convergence together unconditional convergence.
 Unconditional convergence absoluteconvergence, even if N is Banach.
 If N is a Euclidean space, then unconditional convergence absoluteconvergence.
^{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)
...from a set (S) to a uniform space (U)
 Uniform convergence
 Pointwise Cauchyconvergence
 Uniform Cauchyconvergence
Implications are cases of earlier ones, except:
 Uniform convergence both pointwise convergence and uniform Cauchyconvergence.
 Uniform Cauchyconvergence 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 (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 X 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 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 (of partial sum sequence)
 Uniform convergence (of partial sum sequence)
 Pointwise Cauchyconvergence (of partial sum sequence)
 Uniform Cauchyconvergence (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 "absoluteconvergence" means one is referring to convergence of the series of nonnegative functions Σ  g_{k}  in place of Σg_{k}.
 Pointwise absoluteconvergence (pointwise convergence of Σ  g_{k}  )
 Uniform absoluteconvergence (uniform convergence of Σ  g_{k}  )
 Normal convergence^{[1]} (convergence of the series of uniform norms Σ   g_{k}   _{u})
Implications are cases of earlier ones, except:
 Normal convergence uniform absoluteconvergence
...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 absoluteconvergence
 Compact (uniform) absoluteconvergence
 Local normal convergence
 Compact normal convergence
Implications (mostly cases of earlier ones):
 Uniform absoluteconvergence both local uniform absoluteconvergence and compact (uniform) absoluteconvergence.
Normal convergence both local normal convergence and compact normal convergence.
 Local normal convergence local uniform absoluteconvergence.
Compact normal convergence compact (uniform) absoluteconvergence.
 Local uniform absoluteconvergence compact (uniform) absoluteconvergence.
Local normal convergence compact normal convergence
 If X is locally compact:
Local uniform absoluteconvergence compact (uniform) absoluteconvergence.
Local normal convergence compact normal convergence
See also
Categories: Convergence (mathematics)
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