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, 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 is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function.
In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space is continuous. If, in addition, fn 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 x0 ∈ X if for every ε > 0, there exists a δ > 0 such that d, ƒ) < ε for all ƒ ∈ F and all x such that d < 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) < ε for all ƒ ∈ F and all x1, x2 ∈ X such that d < δ.
For comparison, the statement 'all functions ƒ in F are continuous' means that for every ε > 0, every ƒ ∈ F, and every x0 ∈ X, there exists a δ > 0 such that d, ƒ) < ε for all x ∈ X such that d < 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 Ux 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 "equiconituity" 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.

Examples

Suppose that is a topological space and is an additive topological group.
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 operators

Note that every topological vector space 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 operators

Let and be topological vector spaces and be a family of linear operators from into.
Then the following are equivalent:

  1. is equicontinuous;
  2. is equicontinuous at every point of ;
  3. is equicontinuous at some point of ;
  4. is equicontinuous at ;
    • i.e. for every neighborhood of in, there exists a neighborhood of in such that .
  5. for every neighborhood of in, is a neighborhood of in ;
  6. the closure of in is equicontinuous;
    • denotes endowed with the topology of point-wise convergence;
  7. the balanced hull of is equicontinuous;
while if is locally convex then we may add to this list:

  1. the convex hull of is equicontinuous;
  2. the convex balanced hull of is equicontinuous;
while if and are locally convex then we may add to this list:

  1. for every continuous seminorm on, there exists a continuous seminorm on such that for all ;
    • Here, means that for all.
while if is barreled and is locally convex then we may add to this list:

  1. is bounded in ;
  2. is bounded in ;
    • denotes endowed with the topology of bounded convergence.

Characterization of equicontinuous linear functionals

Let be a topological vector space with continuous dual space.
For any subset of, the following are equivalent:

  1. is equicontinuous;
  2. is equicontinuous at the origin;
  3. is equicontinuous at some point of ;
  4. is contained in the polar of some neighborhood of in ;
  5. the polar of is a neighborhood of 0 in ;
  6. the weak* closure of in is equicontinuous;
  7. the balanced hull of is equicontinuous;
  8. the convex hull of is equicontinuous;
  9. the convex balanced hull of is equicontinuous;
while if is normed then we may add to this list:

  1. is a strongly bounded subset of ;
while if is barreled then we may add to this list:

  1. is relatively compact in the weak* topology on ;
  2. is weak* bounded ;
  3. is bounded in the topology of bounded convergence.

Properties of equicontinuous linear maps

The uniform boundedness principle states that a set of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; i.e., for each. The result can be generalized to a case when is locally convex and is a barreled space.

Properties of equicontinuous linear functionals

implies that the weak-* closure of an equicontinuous subset of is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.
If is any locally convex TVS, then the family of all barrels in and the family of all subsets of that are convex, balanced, closed, and bounded in, correspond to each other by polarity.
It follows that a locally convex TVS is barreled if and only if each bounded subset of is equicontinuous.

Equicontinuity and uniform convergence

Let X be a compact Hausdorff space, and equip C with the uniform norm, thus making C a Banach space, hence a metric space. Then Arzelà–Ascoli theorem states that a subset of C 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 Rn are compact if and only if they are closed and bounded.
As a corollary, every uniformly bounded equicontinuous sequence in C contains a subsequence that converges uniformly to a continuous function on X.
In view of Arzelà–Ascoli theorem, a sequence in C 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 converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on X .
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.
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 = 1, and consider the equicontinuous sequence of functions on R defined by ƒn =. Then, ƒn 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 Rn. 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, 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 and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example, ƒn = 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:
;Introduction to uniform spaces
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 .
Every element of is called an entourage.
Uniformities generalize the idea of points that are "-close", meaning that their distance is <.
To clarify this, suppose that is a metric space
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 without 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 to a broader category of topological spaces.
In particular, to topological groups and topological vector spaces.
;A weaker concept is that of even continuity:

Stochastic equicontinuity

Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of random variables, and their convergence.