The purpose of this article is to serve as an annotatedindex 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, 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, 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:
Implications: - Convergence Cauchy-convergence - Cauchy-convergence and convergence of a subsequence together convergence. - U is called "complete" if Cauchy-convergence convergence. Note: A sequence exhibiting Cauchy-convergence is called a cauchy sequence to emphasize that it may not be convergent.
Implications: - Absolute-convergence Cauchy-convergence absolute-convergence of some grouping1. - Therefore: N is Banach 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 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'')
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'')
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 convergence. - "Local" modes of convergence tend to imply "compact" modes of convergence. E.g., Local uniform convergence compact convergence. - If is locally compact, the converses to such tend to hold: Local uniform convergence compact convergence.
Implications : - Uniform absolute-convergence both local uniform absolute-convergence and compact absolute-convergence. Normal convergence both local normal convergence and compact normal convergence. - Local normal convergence local uniform absolute-convergence. Compact normal convergence compact absolute-convergence. - Local uniform absolute-convergence compact absolute-convergence. Local normal convergence compact normal convergence - If X is locally compact: Local uniform absolute-convergence compact absolute-convergence. Local normal convergence compact normal convergence
