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.
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.
Finally, subheadings will always indicate special cases of their super headings.
- 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.
- 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.
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
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.
- "Local" modes of convergence tend to imply "compact" modes of convergence.
compact (uniform) convergence.
is locally compact, the converses to such tend to hold: Local uniform convergence
compact (uniform) convergence.
(In a finite measure space) - Almost uniform convergence
convergence in distribution if μ is a probability measure and the functions are integrable.
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
uniform absolute-convergence Implications are all cases of earlier ones.
Implications (mostly cases of earlier ones): - Uniform absolute-convergence
Compact normal convergence
compact (uniform) absolute-convergence.
compact normal convergence - If X is locally compact: Local uniform absolute-convergence