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| − | A type of convergence of sequences of functions (mappings). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732302.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732303.png" /> is some set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732304.png" /> is a [[Topological space|topological space]]; then pointwise convergence means that for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732305.png" /> the sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732307.png" /> converges in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732308.png" />. An important subclass of the pointwise-convergent sequences for the case of mappings between metric spaces (or, more generally, uniform spaces) is that of the uniformly-convergent sequences (cf. [[Uniform convergence|Uniform convergence]]).
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| | + | A type of convergence of sequences of functions (mappings). Let $f_n : X \rightarrow Y$, $n=1,2,\ldots$ where $X$ is some set and $Y$ is a [[topological space]]; then pointwise convergence means that for any element $x \in X$ the sequence of values $y_n = f_n(x)$, $n=1,2,\ldots$ converges in the space $Y$. The function $f : x \mapsto \lim_n y_n$ is then the '''pointwise limit''' of the sequence $(f_n)$. The definition extends to [[generalized sequence]]s of functions and their values. |
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| | + | An important subclass of the pointwise-convergent sequences for the case of mappings between metric spaces (or, more generally, uniform spaces) is that of the uniformly-convergent sequences (cf. [[Uniform convergence]]). |
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| − | ====Comments====
| + | See also [[Pointwise convergence, topology of]]. |
| − | A base for the topology of pointwise convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p0732309.png" />, the space of continuous mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323011.png" />, is obtained as follows. Take a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323012.png" /> and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323013.png" /> an open subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323014.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323015.png" />; for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323016.png" /> an open basis neighbourhood is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073230/p07323017.png" />. See also [[Pointwise convergence, topology of|Pointwise convergence, topology of]].
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 86 (Translated from Russian)</TD></TR></table>
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Latest revision as of 17:36, 31 December 2016
2020 Mathematics Subject Classification: Primary: 54C35 [MSN][ZBL]
A type of convergence of sequences of functions (mappings). Let $f_n : X \rightarrow Y$, $n=1,2,\ldots$ where $X$ is some set and $Y$ is a topological space; then pointwise convergence means that for any element $x \in X$ the sequence of values $y_n = f_n(x)$, $n=1,2,\ldots$ converges in the space $Y$. The function $f : x \mapsto \lim_n y_n$ is then the pointwise limit of the sequence $(f_n)$. The definition extends to generalized sequences of functions and their values.
An important subclass of the pointwise-convergent sequences for the case of mappings between metric spaces (or, more generally, uniform spaces) is that of the uniformly-convergent sequences (cf. Uniform convergence).
See also Pointwise convergence, topology of.
How to Cite This Entry:
Pointwise convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence&oldid=11463
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article