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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 points $y_n = f_n(x)$, $n=1,2,\ldots$ converges in the space $Y$. 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]]).
  
  
  
 
====Comments====
 
====Comments====
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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A base for the topology of pointwise convergence on $C(X,Y)$, the space of continuous mappings from $X$ to $Y$, is obtained as follows. Take a finite set $K \subset X$ and for each $x \in K$ an open subset $V_x$ in $Y$ containing $f(x)$; for a given $f$ an open basis neighbourhood is: $\{ g \in C(X,Y) : g(x) \in V_x\ \text{for all}\ x \in K \}$. See also [[Pointwise convergence, topology of]].
  
 
====References====
 
====References====
<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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<table>
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<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>
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</table>

Revision as of 18:11, 1 December 2014

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 points $y_n = f_n(x)$, $n=1,2,\ldots$ converges in the space $Y$. 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).


Comments

A base for the topology of pointwise convergence on $C(X,Y)$, the space of continuous mappings from $X$ to $Y$, is obtained as follows. Take a finite set $K \subset X$ and for each $x \in K$ an open subset $V_x$ in $Y$ containing $f(x)$; for a given $f$ an open basis neighbourhood is: $\{ g \in C(X,Y) : g(x) \in V_x\ \text{for all}\ x \in K \}$. See also Pointwise convergence, topology of.

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 86 (Translated from Russian)
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