Difference between revisions of "Pointwise convergence"

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).

How to Cite This Entry:
Pointwise convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence&oldid=40122

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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-{{TEX|done}}
+{{TEX|done}}{{MSC|54C35}}
-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  $Y$ .  The function $f : x \mapsto \lim_n f_n(x) $is then the '''pointwise limit''' of the sequence$ (f_n)$.
+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.
 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====
+See also [[Pointwise convergence, topology of]].
-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====
-<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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Difference between revisions of "Pointwise convergence"

Latest revision as of 17:36, 31 December 2016