Equicontinuity
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus "sequences" of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of "C"("X"), the space of continuous functions on a compact Hausdorff space "X", is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in "C"("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 functions "f" on either metric space or locally compact space is continuous. If, in addition, "f" are holomorphic, 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. Equicontinuity between metric spaces. Let "X" and "Y" be two metric spaces, and "F" a family of functions from "X" to "Y". We shall denote by "d" the respective metrics of these spaces . The family "F" is equicontinuous at a point "x" ∈ "X" if for every ε > 0, there exists a  δ > 0 such that "d"("ƒ"("x"), "ƒ"("x")) < ε for all "ƒ" ∈ "F" and all "x" such that "d"("x", "x") < δ. The family is pointwise equicontinuous if it is equicontinuous at each point of "X". The family "F" is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that "d"("ƒ"("x"), "ƒ"("x")) < ε for all "ƒ" ∈ "F" and all "x", "x" ∈ "X" such that "d"("x", "x") < δ. For comparison , the statement 'all functions "ƒ" in "F" are continuous' means that for every ε > 0, every "ƒ" ∈ "F", and every "x" ∈ "X", there exists a δ > 0 such that "d"("ƒ"("x"), "ƒ"("x")) < ε for all "x" ∈ "X" such that "d"("x", "x") < δ. More generally, when "X" is a topological space, a set "F" of functions from "X" to "Y" is said to be equicontinuous at "x" if for every ε > 0, "x" has a neighborhood "U" such that for all and "ƒ" ∈ "F". This definition usually appears in the context of topological  vector spaces. When "X" is 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 is uniformly continuous , and every finite set of uniformly continuous functions is uniformly equicontinuous. Equicontinuity of maps valued in topological groups. Suppose that is a topological space and is an additive topological group (i.e. a group endowed with a topology making its operations continuous). Topological vector spaces are prominent examples of topological groups and every topological group has an associated canonical uniformity. Note that if is equicontinuous at a point then every map in is continuous at the point. Clearly, every finite set of continuous maps from into is equicontinuous. Equicontinuous linear maps. Because every topological 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 formula_2 of maps of the form formula_3 between two topological vector spaces is said to be formula_4 if for every neighborhood formula_5 of the origin in formula_6 there exists some neighborhood formula_7 of the origin in formula_8 such that formula_9 for all formula_10 If formula_2 is a family of maps and formula_7 is a set then let formula_13 With notation, if formula_7 and formula_5 are sets then formula_16 for all formula_17 if and only if  formula_18 Let formula_8 and formula_6 be topological vector spaces (TVSs) and formula_2 be a family of linear operators from formula_8 into formula_23 Then the following are equivalent: while if formula_6 is locally convex then this list may be extended to include: while if formula_8 and formula_6 are locally convex then this list may be extended to include: while if formula_8 is barreled and formula_6 is locally convex then this list may be extended to include: while if formula_8 and formula_6 are Banach spaces then this list may be extended to include: Characterization of equicontinuous linear functionals. Let formula_8 be a  topological vector space (TVS) over the field formula_32 with continuous dual space formula_33 A family formula_2 of linear functionals on formula_8 is said to be formula_4 if for every neighborhood formula_5 of the origin in formula_32 there exists some neighborhood formula_7 of the origin in formula_8 such that formula_9 for all formula_10 For any subset formula_43 the following are equivalent: while if formula_8 is normed then this list may be extended to include: while if formula_8 is a barreled space then this list may be extended to include: Properties of equicontinuous linear maps. The uniform boundedness principle (also known as  the Banach–Steinhaus theorem) states that a set formula_2 of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, formula_47 for each formula_48 The result can be generalized to a case when formula_6 is locally convex and formula_8 is a barreled space. Properties of equicontinuous linear functionals. Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of formula_51 is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact. If formula_8 is any locally convex TVS, then the family of all barrels in formula_8 and the family of all subsets of  formula_51 that are convex, balanced, closed, and bounded in formula_55 correspond to each other by polarity (with respect to formula_56). It follows that a locally convex TVS formula_8 is barreled if and only if every bounded subset of formula_58 is equicontinuous. Equicontinuity and uniform convergence. Let "X" be a compact Hausdorff space, and equip "C"("X") with the uniform norm, thus making "C"("X") a Banach space, hence a metric space. Then Arzelà–Ascoli theorem states that a subset of "C"("X") is compact if and only if it is closed, uniformly bounded and equicontinuous. This is analogous to the Heine–Borel theorem , which states that subsets of R are compact if and only if they are closed and bounded. As a corollary, every uniformly bounded equicontinuous sequence in "C"("X") contains a subsequence that converges uniformly to a continuous function on "X". In view of Arzelà–Ascoli theorem, a sequence in "C"("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 in "C"("X") converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on "X" (not assumed continuous). 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 of "X"  cannot be relaxed. To see that, consider a compactly supported continuous function "g" on R with "g"(0) = 1, and consider the equicontinuous sequence of functions on R defined by "ƒ"("x") = . Then, "ƒ" converges 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 subset "G" of R. As noted above, it actually converges uniformly on a compact subset of "G" if 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 the mean value theorem or 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 of "G"; thus, continuous on "G". A similar argument can be made when the functions are holomorphic. One can use, for instance, Cauchy's estimate to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example, "ƒ"("x") = converges to a multiple of the discontinuous sign function. Generalizations. Equicontinuity in topological spaces. The most general scenario in which equicontinuity can be defined is for topological spaces whereas "uniform  equicontinuity requires the filter of neighbourhoods of one point to be somehow comparable with the filter of neighbourhood of another point. The latter is most generally done via a uniform structure, giving a uniform space. Appropriate definitions in these cases are as follows: We now briefly describe the basic idea underlying uniformities. The uniformity is a non-empty collection of subsets of where, among many other properties, every , contains the diagonal of (i.e. ). Every element of is called an entourage. Uniformities generalize the idea (taken from metric spaces) of points that are -close" (for ), meaning that their distance is <  . To clarify this, suppose that is a metric space (so the diagonal of is the set ) For any , let denote the set of all pairs of points that are -close. Note that if we were to "forget" that existed then, for any , we would still be able to determine whether or not two points of are -close by using only the sets . In this way, the sets encapsulate all the information necessary to define things such as uniform continuity and uniform convergence with"out" needing any metric. Axiomatizing the most basic properties of these sets leads to the definition of a  uniformity. Indeed, the sets generate the uniformity that is canonically associated with the metric space . 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, to topological groups and topological vector spaces. Stochastic equicontinuity. Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of random variables, and their convergence.