Editing Arzelà–Ascoli theorem (section)

==Generalizations==

===Euclidean spaces===
The Arzelà–Ascoli theorem holds, more generally, if the functions {{math|&thinsp;''f<sub>n</sub>''&thinsp;}} take values in {{mvar|d}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''d''</sup>}}, and the proof is very simple: just apply the {{math|'''R'''}}-valued version of the Arzelà–Ascoli theorem {{mvar|d}} times to extract a subsequence that converges uniformly in the first coordinate, then a sub-subsequence that converges uniformly in the first two coordinates, and so on.  The above examples generalize easily to the case of functions with values in Euclidean space.

=== Compact metric spaces and compact Hausdorff spaces ===
The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary compact [[metric space]]s and, more generally still, [[compact set|compact]] [[Hausdorff space]]s.  Let ''X'' be a compact Hausdorff space, and let ''C''(''X'') be the space of real-valued [[continuous function]]s on ''X''. A subset {{math|'''F''' ⊂ ''C''(''X'')}} is said to be ''equicontinuous'' if for every ''x''&nbsp;∈ ''X'' and every {{math|''ε'' > 0}}, ''x'' has a neighborhood ''U<sub>x</sub>'' such that

:<math>\forall y \in U_x, \forall f \in \mathbf{F} : \qquad |f(y) - f(x)| < \varepsilon.</math>

A set {{math|'''F''' ⊂ ''C''(''X'', '''R''')}} is said to be ''pointwise bounded'' if for every ''x''&nbsp;∈ ''X'',

:<math>\sup \{ | f(x) | : f \in \mathbf{F} \} < \infty.</math>

A version of the Theorem holds also in the space ''C''(''X'') of real-valued continuous functions on a [[compact set|compact]] [[Hausdorff space]] ''X'' {{harv|Dunford|Schwartz|1958|loc=§IV.6.7}}:

:Let ''X'' be a compact Hausdorff space. Then a subset '''F''' of ''C''(''X'') is [[relatively compact]] in the topology induced by the [[uniform norm]] [[if and only if]] it is [[equicontinuous]] and pointwise bounded.

The Arzelà–Ascoli theorem is thus a fundamental result in the study of the algebra of [[continuous functions on a compact Hausdorff space]].

Various generalizations of the above quoted result are possible.  For instance, the functions can assume values in a metric space or (Hausdorff) [[topological vector space]] with only minimal changes to the statement (see, for instance, {{harvtxt|Kelley|Namioka|1982|loc=§8}}, {{harvtxt|Kelley|1991|loc=Chapter 7}}):

:Let ''X'' be a compact Hausdorff space and ''Y'' a metric space. Then {{math|'''F''' ⊂ ''C''(''X'', ''Y'')}} is compact in the [[compact-open topology]] if and only if it is [[equicontinuous]], pointwise [[relatively compact]] and closed.

Here pointwise relatively compact means that for each ''x''&nbsp;∈&nbsp;''X'', the set {{math|'''F'''<sub>''x''</sub> {{=}} {&thinsp;''f''&thinsp;(''x'') : &thinsp;''f''&thinsp; ∈ '''F'''} }}is relatively compact in ''Y''.

In the case that ''Y'' is [[complete metric space|complete]], the proof given above can be generalized in a way that does not rely on the [[separable metric space|separability]] of the domain.  On a [[compact Hausdorff space]] ''X'', for instance, the equicontinuity is used to extract, for each ε&nbsp;=&nbsp;1/''n'', a finite open covering of ''X'' such that the [[oscillation (mathematics)|oscillation]] of any function in the family is less than ε on each [[open set]] in the cover.  The role of the rationals can then be played by a set of points drawn from each open set in each of the countably many covers obtained in this way, and the main part of the proof proceeds exactly as above. A similar argument is used as a part of the proof for the general version which does not assume completeness of ''Y''.

===Functions on non-compact spaces===
The Arzela-Ascoli theorem generalises to functions <math>X \rightarrow Y</math> where <math>X</math> is not compact. Particularly important are cases where <math>X</math> is a [[topological vector space]]. Recall that if <math>X</math>
is a [[topological space]] and <math>Y</math> is a [[uniform space]] (such as any metric space or any [[topological group]], metrisable or not), there is the [[topology of compact convergence]] on the set <math>\mathfrak{F}(X,Y)</math> of functions <math>X \rightarrow Y</math>; it is set up so that a sequence (or more generally a 
[[Filter (set theory)|filter]] or [[Net (mathematics)|net]]) of functions converges if and only if it converges ''uniformly'' on each compact subset of <math>X</math>. Let <math>\mathcal{C}_c(X,Y)</math> be the subspace of
<math>\mathfrak{F}(X,Y)</math> consisting of continuous functions, equipped with the topology of compact convergence.
Then one form of the Arzelà-Ascoli theorem is the following:

:Let <math>X</math> be a topological space, <math>Y</math> a [[Hausdorff space|Hausdorff]] uniform space and <math>H\subset\mathcal{C}_c(X,Y)</math> an [[equicontinuous]] set of continuous functions such that <math>\{h(x) : h \in H\}</math> is [[Relatively compact subspace|relatively compact]] in <math>Y</math> for each <math>x\in X</math>. Then <math>H</math> is relatively compact in <math>\mathcal{C}_c(X,Y)</math>.

This theorem immediately gives the more specialised statements above in cases where <math>X</math> is compact
and the uniform structure of <math>Y</math> is given by a metric. There are a few other variants in terms of
the topology of [[precompact space|precompact]] convergence or other related topologies on 
<math>\mathfrak{F}(X,Y)</math>. It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of <math>X</math> by compact subsets. For details one can consult Bourbaki (1998), Chapter X, § 2, nr 5.

===Non-continuous functions===
Solutions of numerical schemes for parabolic equations are usually piecewise constant, and therefore not continuous, in time. As their jumps nevertheless tend to become small as the time step goes to <math>0</math>, it is possible to establish uniform-in-time convergence properties using a generalisation to non-continuous functions of the classical Arzelà–Ascoli theorem (see e.g. {{harvtxt|Droniou|Eymard|2016|loc=Appendix}}).

Denote by <math>S(X,Y)</math> the space of functions from <math>X</math> to <math>Y</math> endowed with the uniform metric

:<math>d_S(v,w)=\sup_{t\in X}d_Y(v(t),w(t)).</math>

Then we have the following:

:Let <math>X</math> be a compact metric space and <math>Y</math> a complete metric space. Let <math>\{v_n\}_{n\in\mathbb{N}}</math> be a sequence in <math>S(X,Y)</math> such that there exists a function <math>\omega:X\times X\to[0,\infty]</math> and a sequence <math>\{\delta_n\}_{n\in\mathbb{N}}\subset[0,\infty)</math> satisfying
::<math>\lim_{d_X(t,t')\to0}\omega(t,t')=0,\quad\lim_{n\to\infty}\delta_n=0,</math>
::<math>\forall(t,t')\in X\times X,\quad \forall n\in\mathbb{N},\quad d_Y(v_n(t),v_n(t'))\leq \omega(t,t')+\delta_n.</math>
:Assume also that, for all <math>t\in X</math>, <math>\{v_n(t):n\in\mathbb{N}\}</math> is relatively compact in <math>Y</math>. Then <math>\{v_n\}_{n\in\mathbb{N}}</math> is relatively compact in <math>S(X,Y)</math>, and any limit of <math>\{v_n\}_{n\in\mathbb{N}}</math> in this space is in <math>C(X,Y)</math>.