Equicontinuity

Inmathematical analysis,a family of functions isequicontinuousif all the functions arecontinuousand they have equal variation over a givenneighbourhood,in a precise sense described herein. In particular, the concept applies tocountablefamilies, and thussequencesof functions.

Equicontinuity appears in the formulation ofAscoli's theorem,which states that a subset ofC(X), the space ofcontinuous functions on a compact Hausdorff spaceX,is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence inC(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functionsf_non either metric space orlocally compact space^[1]is continuous. If, in addition,f_nareholomorphic,then the limit is also holomorphic.

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.^[2]

Equicontinuity between metric spaces

LetXandYbe twometric spaces,andFa family of functions fromXtoY.We shall denote bydthe respective metrics of these spaces.

The familyFisequicontinuous at a pointx₀∈Xif for every ε > 0, there exists a δ > 0 such thatd(ƒ(x₀),ƒ(x)) < ε for allƒ∈Fand allxsuch thatd(x₀,x) < δ. The family ispointwise equicontinuousif it is equicontinuous at each point ofX.^[3]

The familyFisuniformly equicontinuousif for every ε > 0, there exists a δ > 0 such thatd(ƒ(x₁),ƒ(x₂)) < ε for allƒ∈Fand allx₁,x₂∈Xsuch thatd(x₁,x₂) < δ.^[4]

For comparison, the statement 'all functionsƒinFare continuous' means that for every ε > 0, everyƒ∈F,and everyx₀∈X,there exists a δ > 0 such thatd(ƒ(x₀),ƒ(x)) < ε for allx∈Xsuch thatd(x₀,x) < δ.

Forcontinuity,δ may depend on ε,ƒ,andx₀.
Foruniform continuity,δ may depend on ε andƒ.
Forpointwise equicontinuity,δ may depend on ε andx₀.
Foruniform equicontinuity,δ may depend only on ε.

More generally, whenXis a topological space, a setFof functions fromXtoYis said to be equicontinuous atxif for every ε > 0,xhas a neighborhoodU_xsuch that

d_{Y}(f(y),f(x))<\epsilon

for ally∈U_xandƒ∈F.This definition usually appears in the context oftopological vector spaces.

WhenXis compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces. Used on its own, the term "equicontinuity" may refer to either the pointwise or uniform notion, depending on the context. On a compact space, these notions coincide.

Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous. The closure of an equicontinuous set is again equicontinuous. Every member of a uniformly equicontinuous set of functions isuniformly continuous,and every finite set of uniformly continuous functions is uniformly equicontinuous.

Examples

A set of functions with a commonLipschitz constantis (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant.
Uniform boundedness principlegives a sufficient condition for a set of continuous linear operators to be equicontinuous.
A family of iterates of ananalytic functionis equicontinuous on theFatou set.^[5]^[6]

Counterexamples

The sequence of functions f_n(x) = arctan(nx), is not equicontinuous because the definition is violated at x₀=0.

Equicontinuity of maps valued in topological groups

Suppose that $T$ is a topological space and $Y$ is an additivetopological group(i.e. agroupendowed with a topology making its operations continuous). Topological vector spacesare prominent examples of topological groups and every topological group has an associated canonicaluniformity.

Definition:^[7]A family

H

of maps from

T

into

Y

is said to beequicontinuous at

t \in T

if for every neighborhood

V

of

0

in

Y

,there exists some neighborhood

U

of

t

in

T

such that

h (U) \subseteq h (t) + V

for every

h \in H

.We say that

H

isequicontinuousif it is equicontinuous at every point of

T

.

Note that if $H$ is equicontinuous at a point then every map in $H$ is continuous at the point. Clearly, every finite set of continuous maps from $T$ into $Y$ is equicontinuous.

Equicontinuous linear maps

Because everytopological vector space(TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.

Characterization of equicontinuous linear maps

A family $H$ of maps of the form $X\to Y$ between two topological vector spaces is said to beequicontinuous at a point $x\in X$ if for every neighborhood $V$ of the origin in $Y$ there exists some neighborhood $U$ of the origin in $X$ such that $h(x+U)\subseteq h(x)+V$ for all $h\in H.$

If $H$ is a family of maps and $U$ is a set then let $H(U):=\bigcup _{h\in H}h(U).$ With notation, if $U$ and $V$ are sets then $h(U)\subseteq V$ for all $h\in H$ if and only if $H(U)\subseteq V.$

Let $X$ and $Y$ betopological vector spaces(TVSs) and $H$ be a family of linear operators from $X$ into $Y.$ Then the following are equivalent:

$H$ is equicontinuous;
$H$ is equicontinuous at every point of $X.$
$H$ is equicontinuous at some point of $X.$
$H$ $H$ is equicontinuous at the origin.
- that is, for every neighborhood $V$ of the origin in $Y,$ there exists a neighborhood $U$ of the origin in $X$ such that $H(U)\subseteq V$ (or equivalently, $h(U)\subseteq V$ for every $h\in H$ ).
for every neighborhood $V$ of the origin in $Y,$ $\bigcap _{h\in H}h^{-1}(V)$ is a neighborhood of the origin in $X.$
the closure of $H$ $H$ in $L_{\sigma }(X;Y)$ $L_{\sigma }(X;Y)$ is equicontinuous.
- $L_{\sigma }(X;Y)$ denotes $L(X;Y)$ endowed with the topology of point-wise convergence.
thebalanced hullof $H$ is equicontinuous.

while if $Y$ islocally convexthen this list may be extended to include:

theconvex hullof $H$ is equicontinuous.^[9]
theconvex balanced hullof $H$ is equicontinuous.^[10]^[9]

while if $X$ and $Y$ arelocally convexthen this list may be extended to include:

for every continuousseminorm $q$ $q$ on $Y,$ $Y,$ there exists a continuous seminorm $p$ $p$ on $X$ $X$ such that $q\circ h\leq p$ $q\circ h\leq p$ for all $h\in H.$ $h\in H.$ ^[9]
- Here, $q\circ h\leq p$ means that $q(h(x))\leq p(x)$ for all $x\in X.$

while if $X$ isbarreledand $Y$ is locally convex then this list may be extended to include:

$H$ is bounded in $L_{\sigma }(X;Y)$ ;^[11]
$H$ $H$ is bounded in $L_{b}(X;Y).$ $L_{b}(X;Y).$ ^[11]
- $L_{b}(X;Y)$ denotes $L(X;Y)$ endowed with the topology of bounded convergence (that is, uniform convergence on bounded subsets of $X.$

while if $X$ and $Y$ areBanach spacesthen this list may be extended to include:

$\sup\{\|T\|:T\in H\}<\infty$ (that is, $H$ is uniformly bounded in theoperator norm).

Characterization of equicontinuous linear functionals

Let $X$ be atopological vector space(TVS) over the field $\mathbb {F}$ withcontinuous dual space $X^{\prime }.$ A family $H$ of linear functionals on $X$ is said to beequicontinuous at a point $x\in X$ if for every neighborhood $V$ of the origin in $\mathbb {F}$ there exists some neighborhood $U$ of the origin in $X$ such that $h(x+U)\subseteq h(x)+V$ for all $h\in H.$

For any subset $H\subseteq X^{\prime },$ the following are equivalent:^[9]

$H$ is equicontinuous.
$H$ is equicontinuous at the origin.
$H$ is equicontinuous at some point of $X.$
$H$ is contained in thepolarof some neighborhood of the origin in $X$ ^[10]
the(pre)polarof $H$ is a neighborhood of the origin in $X.$
theweak* closureof $H$ in $X^{\prime }$ is equicontinuous.
thebalanced hullof $H$ is equicontinuous.
theconvex hullof $H$ is equicontinuous.
theconvex balanced hullof $H$ is equicontinuous.^[10]

while if $X$ isnormedthen this list may be extended to include:

$H$ is a strongly bounded subset of $X^{\prime }.$ ^[10]

while if $X$ is abarreled spacethen this list may be extended to include:

$H$ isrelatively compactin theweak* topologyon $X^{\prime }.$ ^[11]
$H$ isweak* bounded(that is, $H$ is $\sigma \left(X^{\prime },X\right)-$ bounded in $X^{\prime }$ ).^[11]
$H$ is bounded in the topology of bounded convergence (that is, $H$ is $b\left(X^{\prime },X\right)-$ bounded in $X^{\prime }$ ).^[11]

Properties of equicontinuous linear maps

Theuniform boundedness principle(also known as the Banach–Steinhaus theorem) states that a set $H$ of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, $\sup _{h\in H}\|h(x)\|<\infty$ for each $x\in X.$ The result can be generalized to a case when $Y$ is locally convex and $X$ is abarreled space.^[12]

Properties of equicontinuous linear functionals

Alaoglu's theoremimplies that the weak-* closure of an equicontinuous subset of $X^{\prime }$ is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.^[13]^[9]

If $X$ is any locally convex TVS, then the family of allbarrelsin $X$ and the family of all subsets of $X^{\prime }$ that are convex, balanced, closed, and bounded in $X_{\sigma }^{\prime },$ correspond to each other by polarity (with respect to $\left\langle X,X^{\#}\right\rangle$ ).^[14] It follows that a locally convex TVS $X$ is barreled if and only if every bounded subset of $X_{\sigma }^{\prime }$ is equicontinuous.^[14]

Theorem—Suppose that $X$ is aseparableTVS. Then every closed equicontinuous subset of $X_{\sigma }^{\prime }$ is a compact metrizable space (under the subspace topology). If in addition $X$ is metrizable then $X_{\sigma }^{\prime }$ is separable.^[14]

Equicontinuity and uniform convergence

LetXbe a compact Hausdorff space, and equipC(X) with theuniform norm,thus makingC(X) aBanach space,hence a metric space. ThenArzelà–Ascoli theoremstates that a subset ofC(X) is compact if and only if it is closed, uniformly bounded and equicontinuous.^[15] This is analogous to theHeine–Borel theorem,which states that subsets ofRⁿare compact if and only if they are closed and bounded.^[16] As a corollary, every uniformly bounded equicontinuous sequence inC(X) contains a subsequence that converges uniformly to a continuous function onX.

In view of Arzelà–Ascoli theorem, a sequence inC(X) converges uniformly if and only if it is equicontinuous and converges pointwise. The hypothesis of the statement can be weakened a bit: a sequence inC(X) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function onX(not assumed continuous).

Proof

Supposef_jis an equicontinuous sequence of continuous functions on a dense subsetDofX. Letε> 0 be given. By equicontinuity, for eachz∈D,there exists a neighborhoodU_zofzsuch that

|f_{j}(x)-f_{j}(z)|<\epsilon /3

for alljandx∈U_z. By denseness and compactness, we can find a finite subsetD′⊂Dsuch thatXis the union ofU_zoverz∈D′.Sincef_jconverges pointwise onD′,there existsN> 0 such that

|f_{j}(z)-f_{k}(z)|<\epsilon /3

wheneverz∈D′andj,k>N.It follows that

\sup _{X}|f_{j}-f_{k}|<\epsilon

for allj,k>N.In fact, ifx∈X,thenx∈U_zfor somez∈D′and so we get:

|f_{j}(x)-f_{k}(x)|\leq |f_{j}(x)-f_{j}(z)|+|f_{j}(z)-f_{k}(z)|+|f_{k}(z)-f_{k}(x)|<\epsilon

.

Hence,f_jis Cauchy inC(X) and thus converges by completeness.

This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous. (See below for an example.) In the above, the hypothesis of compactness ofX cannot be relaxed. To see that, consider a compactly supported continuous functiongonRwithg(0) = 1, and consider the equicontinuous sequence of functions {ƒ_n} onRdefined byƒ_n(x) =g(x−n).Then,ƒ_nconverges pointwise to 0 but does not converge uniformly to 0.

This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subsetGofRⁿ.As noted above, it actually converges uniformly on a compact subset ofGif it is equicontinuous on the compact set. In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then themean value theoremor some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset ofG;thus, continuous onG.A similar argument can be made when the functions are holomorphic. One can use, for instance,Cauchy's estimateto show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example,ƒ_n(x) =arctann xconverges to a multiple of the discontinuoussign function.

Generalizations

Equicontinuity in topological spaces

The most general scenario in which equicontinuity can be defined is fortopological spaceswhereasuniformequicontinuity requires thefilterof neighbourhoods of one point to be somehow comparable with the filter of neighbourhood of another point. The latter is most generally done via auniform structure,giving auniform space.Appropriate definitions in these cases are as follows:

A setAof functions continuous between two topological spacesXandYistopologically equicontinuous at the pointsx∈Xandy∈Yif for any open setOabouty,there are neighborhoodsUofxandVofysuch that for everyf∈A,if the intersection off[U] andVis nonempty,f[U] ⊆O.ThenAis said to betopologically equicontinuous atx∈Xif it is topologically equicontinuous atxandyfor eachy∈Y.Finally,Aisequicontinuousif it is equicontinuous atxfor all pointsx∈X.

A setAof continuous functions between two uniform spacesXandYisuniformly equicontinuousif for every elementWof the uniformity onY,the set

{ (u,v) ∈X × X:for allf∈A.(f(u),f(v)) ∈W}

is a member of the uniformity onX

Introduction to uniform spaces

We now briefly describe the basic idea underlying uniformities.

The uniformity $𝒱$ is a non-empty collection of subsets of $Y \times Y$ where, among many other properties, every $V \in 𝒱$ , $V$ contains the diagonal of $Y$ (i.e. ${(y, y) \in Y}$ ). Every element of $𝒱$ is called anentourage.

Uniformities generalize the idea (taken frommetric spaces) of points that are " $r$ -close "(for $r > 0$ ), meaning that their distance is < $r$ . To clarify this, suppose that $(Y, d)$ is a metric space (so the diagonal of $Y$ is the set ${(y, z) \in Y \times Y : d (y, z) = 0}$ ) For any $r > 0$ ,let

U r = {(y, z) \in Y \times Y : d (y, z) < r}

denote the set of all pairs of points that are $r$ -close. Note that if we were to "forget" that $d$ existed then, for any $r > 0$ ,we would still be able to determine whether or not two points of $Y$ are $r$ -close by using only the sets $U r$ . In this way, the sets $U r$ encapsulate all the information necessary to define things such asuniform continuityanduniform convergencewithoutneeding any metric. Axiomatizing the most basic properties of these sets leads to the definition of auniformity. Indeed, the sets $U r$ generate the uniformity that is canonically associated with the metric space $(Y, d)$ .

The benefit of this generalization is that we may now extend some important definitions that make sense for metric spaces (e.g.completeness) to a broader category of topological spaces. In particular, totopological groupsandtopological vector spaces.

A weaker concept is that of even continuity

A setAof continuous functions between two topological spacesXandYis said to beevenly continuous atx∈Xandy∈Yif given any open setOcontainingythere are neighborhoodsUofxandVofysuch thatf[U] ⊆Owheneverf(x) ∈V.It isevenly continuous atxif it is evenly continuous atxandyfor everyy∈Y,andevenly continuousif it is evenly continuous atxfor everyx∈X.

Stochastic equicontinuity

Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions ofrandom variables,and theirconvergence.^[17]

Notes

^More generally, on anycompactly generated space;e.g., afirst-countable space.
^Rudin 1991,p. 44 §2.5.
^Reed & Simon (1980),p. 29;Rudin (1987),p. 245
^Reed & Simon (1980),p. 29
^Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet: Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000;ISBN 0-387-95151-2,ISBN 978-0-387-95151-5;page 49
^Joseph H. Silverman: The arithmetic of dynamical systems. Springer, 2007.ISBN 0-387-69903-1,ISBN 978-0-387-69903-5;page 22
^Narici & Beckenstein 2011,pp. 133–136.
^Rudin 1991,p. 44 Theorem 2.4.
^^a ^b ^c ^d ^eNarici & Beckenstein 2011,pp. 225–273.
^^a ^b ^c ^dTrèves 2006,pp. 335–345.
^^a ^b ^c ^d ^eTrèves 2006,pp. 346–350.
^Schaefer 1966,Theorem 4.2.
^Schaefer 1966,Corollary 4.3.
^^a ^b ^cSchaefer & Wolff 1999,pp. 123–128.
^Rudin 1991,p. 394 Appendix A5.
^Rudin 1991,p. 18 Theorem 1.23.
^de Jong, Robert M. (1993). "Stochastic Equicontinuity for Mi xing Processes".Asymptotic Theory of Expanding Parameter Space Methods and Data Dependence in Econometrics.Amsterdam. pp. 53–72.ISBN 90-5170-227-2.{{cite book}}:CS1 maint: location missing publisher (link)

References

"Equicontinuity",Encyclopedia of Mathematics,EMS Press,2001 [1994]
Reed, Michael;Simon, Barry(1980),Functional Analysis(revised and enlarged ed.), Boston, MA:Academic Press,ISBN 978-0-12-585050-6.
Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces.Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.

Rudin, Walter(1991).Functional Analysis.International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 978-0-07-054236-5.OCLC 21163277.

Rudin, Walter(1987),Real and Complex Analysis(3rd ed.), New York:McGraw-Hill.
Schaefer, Helmut H.(1966),Topological vector spaces,New York: The Macmillan Company
Schaefer, Helmut H.;Wolff, Manfred P. (1999).Topological Vector Spaces.GTM.Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
Trèves, François(2006) [1967].Topological Vector Spaces, Distributions and Kernels.Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.

[1] More generally, on anycompactly generated space;e.g., afirst-countable space.

[FOOTNOTERudin199144_§2.5-2] Rudin 1991,p. 44 §2.5.

[RS29-3] Reed & Simon (1980),p. 29;Rudin (1987),p. 245

[4] Reed & Simon (1980),p. 29

[5] Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet: Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000;ISBN 0-387-95151-2,ISBN 978-0-387-95151-5;page 49

[6] Joseph H. Silverman: The arithmetic of dynamical systems. Springer, 2007.ISBN 0-387-69903-1,ISBN 978-0-387-69903-5;page 22

[FOOTNOTENariciBeckenstein2011133–136-7] Narici & Beckenstein 2011,pp. 133–136.

[FOOTNOTERudin199144_Theorem_2.4-8] Rudin 1991,p. 44 Theorem 2.4.

[FOOTNOTENariciBeckenstein2011225–273-9] Narici & Beckenstein 2011,pp. 225–273.

[FOOTNOTETrèves2006335–345-10] Trèves 2006,pp. 335–345.

[FOOTNOTETrèves2006346–350-11] Trèves 2006,pp. 346–350.

[FOOTNOTESchaefer1966Theorem_4.2-12] Schaefer 1966,Theorem 4.2.

[FOOTNOTESchaefer1966Corollary_4.3-13] Schaefer 1966,Corollary 4.3.

[FOOTNOTESchaeferWolff1999123–128-14] Schaefer & Wolff 1999,pp. 123–128.

[FOOTNOTERudin1991394_Appendix_A5-15] Rudin 1991,p. 394 Appendix A5.

[FOOTNOTERudin199118_Theorem_1.23-16] Rudin 1991,p. 18 Theorem 1.23.

[17] Jong, Robert M. (1993). "Stochastic Equicontinuity for Mi xing Processes".Asymptotic Theory of Expanding Parameter Space Methods and Data Dependence in Econometrics.Amsterdam. pp. 53–72.ISBN 90-5170-227-2.{{cite book}}:CS1 maint: location missing publisher (link)

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