Namespaces
Variants
Actions

Difference between revisions of "Equicontinuity"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''of a set of functions''
 
''of a set of functions''
  
An idea closely connected with the concept of compactness of a set of continuous functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359902.png" /> be compact metric spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359903.png" /> be the set of continuous mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359904.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359905.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359906.png" /> is called equicontinuous if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359907.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359908.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e0359909.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599012.png" />. Equicontinuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599013.png" /> is equivalent to the relative compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599015.png" />, equipped with the metric
+
An idea closely connected with the concept of compactness of a set of continuous functions. Let $X$ and $Y$ be compact metric spaces and let $C(X,Y)$ be the set of continuous mappings of $X$ into $Y$. A set $D\subset C(X,Y)$ is called equicontinuous if for any $\epsilon>0$ there is a $\delta>0$ such that $\rho_X(x_1,x_2)\leq\delta$ implies $\rho_Y(f(x_1),f(x_2))\leq\epsilon$ for all $x_1,x_2\in X$, $f\in D$. Equicontinuity of $D$ is equivalent to the relative compactness of $D$ in $C(X,Y)$, equipped with the metric
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035990/e03599016.png" /></td> </tr></table>
+
$$\rho(f,g)=\max_{x\in X}\rho_Y(f(x),g(x));$$
  
 
this is the content of the Arzelà–Ascoli theorem. The idea of equicontinuity can be transferred to uniform spaces.
 
this is the content of the Arzelà–Ascoli theorem. The idea of equicontinuity can be transferred to uniform spaces.

Latest revision as of 08:22, 27 April 2014

of a set of functions

An idea closely connected with the concept of compactness of a set of continuous functions. Let $X$ and $Y$ be compact metric spaces and let $C(X,Y)$ be the set of continuous mappings of $X$ into $Y$. A set $D\subset C(X,Y)$ is called equicontinuous if for any $\epsilon>0$ there is a $\delta>0$ such that $\rho_X(x_1,x_2)\leq\delta$ implies $\rho_Y(f(x_1),f(x_2))\leq\epsilon$ for all $x_1,x_2\in X$, $f\in D$. Equicontinuity of $D$ is equivalent to the relative compactness of $D$ in $C(X,Y)$, equipped with the metric

$$\rho(f,g)=\max_{x\in X}\rho_Y(f(x),g(x));$$

this is the content of the Arzelà–Ascoli theorem. The idea of equicontinuity can be transferred to uniform spaces.

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)


Comments

References

[a1] J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French)
How to Cite This Entry:
Equicontinuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equicontinuity&oldid=17759
This article was adapted from an original article by E.M. Semenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article