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.
Contents
- A sequence of elements an in a topological space Y
- in a uniform space U
- A series of elements bk in a TAG G
- in a normed space N
- A sequence of functions fn from a set S to a topological space Y
- from a set S to a uniform space U
- from a topological space X to a uniform space U
- from a measure space S to the complex numbers C
- A series of functions gk from a set S to a TAG G
- from a set S to a normed space N
- from a topological space X to a TAG G
- from a topological space X to a normed space N
- References
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)
...in a uniform space (U)
Implications:
- Convergence
- Cauchy-convergence and convergence of a subsequence together
- U is called "complete" if Cauchy-convergence (for nets)
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)
Implications:
- Unconditional convergence
...in a normed space (N)
Implications:
- Absolute-convergence
- Therefore: N is Banach (complete) if absolute-convergence
- Absolute-convergence and convergence together
- Unconditional convergence
- If N is a Euclidean space, then unconditional 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)
...from a set (S) to a uniform space (U)
Implications are cases of earlier ones, except:
- Uniform convergence
- Uniform Cauchy-convergence and pointwise convergence of a subsequence
...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:
Implications:
- "Global" modes of convergence imply the corresponding "local" and "compact" modes of convergence. E.g.:
Uniform convergence
- "Local" modes of convergence tend to imply "compact" modes of convergence. E.g.,
Local uniform convergence
- If
Local uniform convergence
...from a measure space (S,μ) to the complex numbers (C)
Implications:
- Pointwise convergence
- Uniform convergence
- Almost everywhere convergence
- Almost uniform convergence
- Lp convergence
- Convergence in measure
A series of functions Σgk from a set (S) to a TAG (G)
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
Implications are cases of earlier ones, except:
- Normal convergence
...from a topological space (X) to a TAG (G)
Implications are all cases of earlier ones.
...from a topological space (X) to a normed space (N)
Implications (mostly cases of earlier ones):
- Uniform absolute-convergence
Normal convergence
- Local normal convergence
Compact normal convergence
- Local uniform absolute-convergence
Local normal convergence
- If X is locally compact:
Local uniform absolute-convergence
Local normal convergence