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 | + | 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]]). | 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]]). | ||
Revision as of 17:31, 31 December 2016
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).
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) |
Pointwise convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence&oldid=40122