187:), which in turn implies absolute convergence of some grouping (not reordering). The sequence of partial sums obtained by grouping is a subsequence of the partial sums of the original series. The convergence of each absolutely convergent series is an equivalent condition for a normed vector space to be
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Pointwise convergence implies pointwise Cauchy convergence, and the converse holds if the space in which the functions take their values is complete. Uniform convergence implies pointwise convergence and uniform Cauchy convergence. Uniform Cauchy convergence and pointwise convergence of a subsequence
282:(even in the weakest sense: every point has compact neighborhood), then local uniform convergence is equivalent to compact (uniform) convergence. Roughly speaking, this is because "local" and "compact" connote the same thing.
194:
Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general, even if the space is Banach, although the implication holds in
271:) may be defined. "Compact convergence" is always short for "compact uniform convergence," since "compact pointwise convergence" would mean the same thing as "pointwise convergence" (points are always compact).
240:. It is defined as convergence of the sequence of values of the functions at every point. If the functions take their values in a uniform space, then one can define pointwise Cauchy convergence,
479:, then several modes of convergence that depend on measure-theoretic, rather than solely topological properties, arise. This includes pointwise convergence almost-everywhere, convergence in
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are spaces in which Cauchy filters may be defined. Convergence implies "Cauchy convergence", and Cauchy convergence, together with the existence of a convergent
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is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of
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Uniform convergence implies both local uniform convergence and compact convergence, since both are local notions while uniform convergence is global. If
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imply uniform convergence of the sequence, and if the codomain is complete, then uniform Cauchy convergence implies uniform convergence.
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The most basic type of convergence for a sequence of functions (in particular, it does not assume any topological structure on the
496:
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Pointwise and uniform convergence of series of functions are defined in terms of convergence of the sequence of partial sums.
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in terms of the partial sums of the series. If, in addition, the functions take values in a normed linear space, then
431:, pointwise absolute convergence implies pointwise convergence, and normal convergence implies uniform convergence.
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Normal convergence implies both local normal convergence and compact normal convergence. And if the domain is
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is defined as convergence of the sequence of partial sums. An important concept when considering series is
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461:(even in the weakest sense), then local normal convergence implies compact normal convergence.
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520:
505: – Use of filters to describe and characterize all basic topological notions and results.
297:, absolute convergence refers to convergence of the series of positive, real-valued functions
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499: – Notions of probabilistic convergence, applied to estimation and asymptotic analysis
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If the domain of the functions is a topological space and the codomain is a uniform space,
183:). Absolute convergence implies Cauchy convergence of the sequence of partial sums (by the
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are a generalization of sequences that are useful in spaces which are not first countable.
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of metric spaces, and its generalizations is defined in terms of Cauchy sequences.
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is convergence of the series of non-negative real numbers obtained by taking the
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337:. "Pointwise absolute convergence" is then simply pointwise convergence of
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Each of the following objects is a special case of the types preceding it:
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For functions defined on a topological space, one can define (as above)
483:-mean and convergence in measure. These are of particular interest in
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Convergence of sequence of functions on a topological space
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of each function in the series (uniform convergence of
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286:Series of functions on a topological abelian group
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120:Series of elements in a topological abelian group
100:. Cauchy nets and filters are generalizations to
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93:further generalize the concept of convergence.
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465:Functions defined on a measure space
112:implies convergence. The concept of
527:Topologies on spaces of linear maps
23:, there are many senses in which a
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293:For functions taking values in a
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217:{\displaystyle \mathbb {R} ^{d}}
16:Property of a sequence or series
497:Convergence of random variables
471:Convergence of random variables
259:(i.e. uniform convergence on a
73:Elements of a topological space
475:If one considers sequences of
420:{\displaystyle \Sigma |g_{k}|}
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370:{\displaystyle \Sigma |g_{k}|}
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176:{\displaystyle \Sigma |b_{k}|}
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143:as convergence of the series (
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440:compact (uniform) convergence
265:compact (uniform) convergence
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448:compact normal convergence
246:uniform Cauchy convergence
547:Convergence (mathematics)
450:(absolute convergence on
436:local uniform convergence
385:uniform (i.e. "sup") norm
257:local uniform convergence
134:unconditional convergence
126:topological abelian group
55:topological abelian group
444:local normal convergence
104:. Even more generally,
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83:first-countable spaces
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238:pointwise convergence
236:of the functions) is
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477:measurable functions
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141:absolute convergence
69:is a uniform space.
33:modes of convergence
509:Limit of a sequence
503:Filters in topology
295:normed linear space
263:of each point) and
242:uniform convergence
185:triangle inequality
128:, convergence of a
485:probability theory
454:) can be defined.
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381:Normal convergence
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191:(i.e.: complete).
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47:topological spaces
521:Net (mathematics)
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59:normed spaces
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452:compact sets
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261:neighborhood
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114:completeness
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67:metric space
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110:subsequence
21:mathematics
536:Categories
395:Σ
345:Σ
305:Σ
151:Σ
79:sequences
542:Topology
491:See also
25:sequence
91:Filters
427:). In
244:, and
234:domain
189:Banach
130:series
35:, see
29:series
124:In a
27:or a
438:and
87:Nets
43:sets
278:is
81:in
19:In
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