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Difference between revisions of "Pointwise 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]]).
 
  
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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====
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See also [[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]].
 
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====References====
 
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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>
 

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=35248
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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