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.

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Guide to this index. To avoid excessive verbiage, note that each of the following types objects is a special case of types preceding it: sets, topological spaces, uniform spaces, TAGs (topological abelians 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:

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)

• Cauchy-convergence

Implications:

• Convergence $\Rightarrow$ Cauchy-convergence
• Cauchy-convergence and convergence of a subsequence together $\Rightarrow$ convergence.
• U is called "complete" if Cauchy-convergence (for nets) $\Rightarrow$ convergence.

Note: A sequence exhibiting Cauchy-convergence is called a cauchy sequence to emphasize that it may not be convergent.

A series of elements Σb_k in a TAG (G)

• Cauchy-convergence (of partial sum sequence)
• Convergence (of partial sum sequence)
• Unconditional convergence

Implications: Unconditional convergence $\Rightarrow$ convergence (by definition).

...in a Normed Space (N)

• Absolute-convergence (convergence of $\sum |b_{k}|$)

Implications:

• Absolute-convergence $\Rightarrow$ Cauchy-convergence $\Rightarrow$ absolute-convergence of some grouping¹.
• Therefore: N is Banach (complete) iff absolute-convergence $\Rightarrow$ convergence.
• Absolute-convergence and convergence together $\Rightarrow$ unconditional convergence.
• Unconditional convergence $\not \Rightarrow$ absolute-convergence, even if N is Banach.
• If N is a Euclidean space, then unconditional convergence $\equiv$ 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 {f_n} from a Set (S) to a Topological Space (Y)

• Pointwise convergence

...from a Set (S) to a Uniform Space (U)

• Pointwise Cauchy-convergence
• Uniform convergence
• Uniform Cauchy-convergence

Implications:

• Pointwise convergence $\Rightarrow$ pointwise Cauchy-convergence, and conversely if U is complete.
• Uniform convergence $\Rightarrow$ both pointwise convergence and uniform Cauchy-convergence.
• Uniform Cauchy-convergence and pointwise convergence of a subsequence $\Rightarrow$ uniform convergence.
• If U is complete, then uniform Cauchy-convergence $\Rightarrow$ uniform convergence.

...from a Topological Space (X) to a Uniform Space (U)

• Local uniform convergence (i.e. uniform convergence on a neighborhood of each point)
• Compact (uniform) convergence (i.e. uniform convergence on all compact subsets)

Note: "Compact convergence" is always short for "compact uniform convergence," since "compact pointwise convergence" would mean the same thing as "pointwise convergence" (points are compact).

Implications:

• Uniform convergence $\Rightarrow$ both local uniform convergence and compact convergence.
• If X is locally compact (even in the weakest sense), then local uniform convergence $\equiv$ compact (uniform) convergence. Roughly speaking, this is because because "local" and "compact" connote the same thing.

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)

...from a Set (S) to a Normed Space (N)

Generally, replacing "convergence" by "absololute-convergence" means one is referring to convergence of the series of positive-valued functions $\Sigma |g_{k}|$ in place of $\Sigma g_{k}$. Examples:

• Pointwise absolute-convergence (pointwise convergence of $\Sigma |g_{k}|$)
• Normal convergence^[1] = uniform absolute-convergence (uniform convergence of $\Sigma |g_{k}|$)
• etc.

Implications: In the above and most senses, if N is Banach, then "xxx" absolute-convergence implies "xxx" convergence.

...from a Topological Space (X) to a TAG (G)

• Local uniform convergence (of partial sum sequence)
• Compact (uniform) convergence (of partial sum sequence)

Implications: same as for sequences of functions from a topological space to a uniform space.

...from a Topological Space (X) to a Normed Space (N)

• Local normal convergence = local uniform absolute-convergence
• Compact normal convergence = compact (uniform) absolute-convergence

Implications (notable consequences of 4.1 and 4.2):

• Normal convergence $\Rightarrow$ both local normal convergence and compact normal convergence.
• If X is locally compact (even in the weakest sense), then local normal convergence $\equiv$ compact normal convergence.