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Difference between revisions of "Quasi-uniform convergence"

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A generalization of [[Uniform convergence|uniform convergence]]. A sequence of mappings $\{f_n\}$ from a topological space $X$ into a metric space $Y$ converging pointwise to a mapping $f$ is called quasi-uniformly convergent if for any $\epsilon>0$ and any positive integer $N$ there exist a countable open covering $\{\Gamma_0,\Gamma_1,\ldots\}$ of $X$ and a sequence $n_0,n_1,\ldots$ of positive integers greater than $N$ such that $\rho(f(x),f_{n_k}(x))<\epsilon$ for every $x\in\Gamma_k$. Uniform convergence implies quasi-uniform convergence. For sequences of continuous functions, quasi-uniform convergence is a necessary and sufficient condition for the limit function to be continuous (the Arzelà–Aleksandrov theorem).
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A generalization of [[uniform convergence]]. A sequence of mappings $\{f_n\}$ from a topological space $X$ into a metric space $Y$ converging pointwise to a mapping $f$ is called quasi-uniformly convergent if for any $\epsilon>0$ and any positive integer $N$ there exist a countable open covering $\{\Gamma_0,\Gamma_1,\ldots\}$ of $X$ and a sequence $n_0,n_1,\ldots$ of positive integers greater than $N$ such that $\rho(f(x),f_{n_k}(x))<\epsilon$ for every $x\in\Gamma_k$. Uniform convergence implies quasi-uniform convergence. For sequences of continuous functions, quasi-uniform convergence is a necessary and sufficient condition for the limit function to be continuous (the Arzelà–Aleksandrov theorem).
  
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR>
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[[Category:Functional analysis]]

Latest revision as of 19:31, 9 November 2014

A generalization of uniform convergence. A sequence of mappings $\{f_n\}$ from a topological space $X$ into a metric space $Y$ converging pointwise to a mapping $f$ is called quasi-uniformly convergent if for any $\epsilon>0$ and any positive integer $N$ there exist a countable open covering $\{\Gamma_0,\Gamma_1,\ldots\}$ of $X$ and a sequence $n_0,n_1,\ldots$ of positive integers greater than $N$ such that $\rho(f(x),f_{n_k}(x))<\epsilon$ for every $x\in\Gamma_k$. Uniform convergence implies quasi-uniform convergence. For sequences of continuous functions, quasi-uniform convergence is a necessary and sufficient condition for the limit function to be continuous (the Arzelà–Aleksandrov theorem).

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
How to Cite This Entry:
Quasi-uniform convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-uniform_convergence&oldid=34444
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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