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 functionsfnon either metric space orlocally compact space[1]is continuous. If, in addition,fnareholomorphic,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
[edit]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 neighborhoodUxsuch that
for ally∈Uxandƒ∈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
[edit]- 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
[edit]- The sequence of functions fn(x) = arctan(nx), is not equicontinuous because the definition is violated at x0=0.
Equicontinuity of maps valued in topological groups
[edit]Suppose thatTis a topological space andYis 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 familyHof maps fromTintoYis said to beequicontinuous att∈Tif for every neighborhoodVof0inY,there exists some neighborhoodUoftinTsuch thath(U) ⊆h(t) +Vfor everyh∈H.We say thatHisequicontinuousif it is equicontinuous at every point ofT.
Note that ifHis equicontinuous at a point then every map inHis continuous at the point. Clearly, every finite set of continuous maps fromTintoYis equicontinuous.
Equicontinuous linear maps
[edit]
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
[edit]A familyof maps of the formbetween two topological vector spaces is said to beequicontinuous at a pointif for every neighborhoodof the origin inthere exists some neighborhoodof the origin insuch thatfor all
Ifis a family of maps andis a set then letWith notation, ifandare sets thenfor allif and only if
Letandbetopological vector spaces(TVSs) andbe a family of linear operators frominto Then the following are equivalent:
- is equicontinuous;
- is equicontinuous at every point of
- is equicontinuous at some point of
- is equicontinuous at the origin.
- that is, for every neighborhoodof the origin inthere exists a neighborhoodof the origin insuch that(or equivalently,for every). [8]
- for every neighborhoodof the origin inis a neighborhood of the origin in
- the closure ofinis equicontinuous.
- denotesendowed with the topology of point-wise convergence.
- thebalanced hullofis equicontinuous.
while ifislocally convexthen this list may be extended to include:
- theconvex hullofis equicontinuous.[9]
- theconvex balanced hullofis equicontinuous.[10][9]
while ifandarelocally convexthen this list may be extended to include:
- for every continuousseminormonthere exists a continuous seminormonsuch thatfor all[9]
- Here,means thatfor all
while ifisbarreledandis locally convex then this list may be extended to include:
- is bounded in;[11]
- is bounded in[11]
- denotesendowed with the topology of bounded convergence (that is, uniform convergence on bounded subsets of
while ifandareBanach spacesthen this list may be extended to include:
- (that is,is uniformly bounded in theoperator norm).
Characterization of equicontinuous linear functionals
[edit]
Letbe atopological vector space(TVS) over the fieldwithcontinuous dual space A familyof linear functionals onis said to beequicontinuous at a pointif for every neighborhoodof the origin inthere exists some neighborhoodof the origin insuch thatfor all
For any subsetthe following are equivalent:[9]
- is equicontinuous.
- is equicontinuous at the origin.
- is equicontinuous at some point of
- is contained in thepolarof some neighborhood of the origin in[10]
- the(pre)polarofis a neighborhood of the origin in
- theweak* closureofinis equicontinuous.
- thebalanced hullofis equicontinuous.
- theconvex hullofis equicontinuous.
- theconvex balanced hullofis equicontinuous.[10]
while ifisnormedthen this list may be extended to include:
- is a strongly bounded subset of[10]
while ifis abarreled spacethen this list may be extended to include:
- isrelatively compactin theweak* topologyon[11]
- isweak* bounded(that is,isbounded in).[11]
- is bounded in the topology of bounded convergence (that is,isbounded in).[11]
Properties of equicontinuous linear maps
[edit]Theuniform boundedness principle(also known as the Banach–Steinhaus theorem) states that a setof linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is,for eachThe result can be generalized to a case whenis locally convex andis abarreled space.[12]
Properties of equicontinuous linear functionals
[edit]Alaoglu's theoremimplies that the weak-* closure of an equicontinuous subset ofis weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.[13][9]
Ifis any locally convex TVS, then the family of allbarrelsinand the family of all subsets ofthat are convex, balanced, closed, and bounded incorrespond to each other by polarity (with respect to).[14] It follows that a locally convex TVSis barreled if and only if every bounded subset ofis equicontinuous.[14]
Theorem—Suppose thatis aseparableTVS. Then every closed equicontinuous subset ofis a compact metrizable space (under the subspace topology). If in additionis metrizable thenis separable.[14]
Equicontinuity and uniform convergence
[edit]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).
Supposefjis an equicontinuous sequence of continuous functions on a dense subsetDofX. Letε> 0 be given. By equicontinuity, for eachz∈D,there exists a neighborhoodUzofzsuch that
for alljandx∈Uz. By denseness and compactness, we can find a finite subsetD′⊂Dsuch thatXis the union ofUzoverz∈D′.Sincefjconverges pointwise onD′,there existsN> 0 such that
wheneverz∈D′andj,k>N.It follows that
for allj,k>N.In fact, ifx∈X,thenx∈Uzfor somez∈D′and so we get:
- .
Hence,fjis 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 {ƒn} onRdefined byƒn(x) =g(x−n).Then,ƒnconverges 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,ƒn(x) =arctann xconverges to a multiple of the discontinuoussign function.
Generalizations
[edit]Equicontinuity in topological spaces
[edit]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 uniformity𝒱is a non-empty collection of subsets ofY×Ywhere, among many other properties, everyV∈ 𝒱,Vcontains the diagonal ofY(i.e.{(y,y) ∈Y}). Every element of𝒱is called anentourage.
Uniformities generalize the idea (taken frommetric spaces) of points that are "r-close "(forr> 0), meaning that their distance is <r. To clarify this, suppose that(Y,d)is a metric space (so the diagonal ofYis the set{(y,z) ∈Y×Y:d(y,z) = 0}) For anyr> 0,let
- Ur= {(y,z) ∈Y×Y:d(y,z) <r}
denote the set of all pairs of points that arer-close. Note that if we were to "forget" thatdexisted then, for anyr> 0,we would still be able to determine whether or not two points ofYarer-close by using only the setsUr. In this way, the setsUrencapsulate 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 setsUrgenerate 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
[edit]Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions ofrandom variables,and theirconvergence.[17]
See also
[edit]- Absolute continuity– Form of continuity for functions
- Classification of discontinuities– Mathematical analysis of discontinuous points
- Coarse function
- Continuous function– Mathematical function with no sudden changes
- Continuous function (set theory)– sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stages
- Continuous stochastic process– Stochastic process that is a continuous function of time or index parameter
- Dini continuity
- Direction-preserving function- an analogue of a continuous function in discrete spaces.
- Microcontinuity– Mathematical term
- Normal function– Function of ordinals in mathematics
- Piecewise– Function defined by multiple sub-functions
- Symmetrically continuous function
- Uniform continuity– Uniform restraint of the change in functions
Notes
[edit]- ^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;ISBN0-387-95151-2,ISBN978-0-387-95151-5;page 49
- ^Joseph H. Silverman: The arithmetic of dynamical systems. Springer, 2007.ISBN0-387-69903-1,ISBN978-0-387-69903-5;page 22
- ^Narici & Beckenstein 2011,pp. 133–136.
- ^Rudin 1991,p. 44 Theorem 2.4.
- ^abcdeNarici & Beckenstein 2011,pp. 225–273.
- ^abcdTrèves 2006,pp. 335–345.
- ^abcdeTrèves 2006,pp. 346–350.
- ^Schaefer 1966,Theorem 4.2.
- ^Schaefer 1966,Corollary 4.3.
- ^abcSchaefer & 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 Mixing Processes".Asymptotic Theory of Expanding Parameter Space Methods and Data Dependence in Econometrics.Amsterdam. pp. 53–72.ISBN90-5170-227-2.
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References
[edit]- "Equicontinuity",Encyclopedia of Mathematics,EMS Press,2001 [1994]
- Reed, Michael;Simon, Barry(1980),Functional Analysis(revised and enlarged ed.), Boston, MA:Academic Press,ISBN978-0-12-585050-6.
- Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces.Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN978-1584888666.OCLC144216834.
- Rudin, Walter(1991).Functional Analysis.International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN978-0-07-054236-5.OCLC21163277.
- 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.ISBN978-1-4612-7155-0.OCLC840278135.
- Trèves, François(2006) [1967].Topological Vector Spaces, Distributions and Kernels.Mineola, N.Y.: Dover Publications.ISBN978-0-486-45352-1.OCLC853623322.