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 a metric space or a 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.
  • For continuity, δ may depend on ε, ƒ, and x0.
  • For uniform continuity, δ may depend on ε and ƒ.
  • For pointwise equicontinuity, δ may depend on ε and x0.
  • For uniform equicontinuity, δ may depend only on ε.
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 "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.

Examples

  • A set of functions with a common Lipschitz constant is equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant.
  • Uniform boundedness principle gives a sufficient condition for a set of continuous linear operators to be equicontinuous.
  • A family of iterates of an analytic function is equicontinuous on the Fatou set.

    Counterexamples

  • The set of all Lipschitz-continuous functions is not equicontinuous, as the maximal Lipschitz-constant is unbounded.
  • The sequence of functions fn = arctan, is not equicontinuous because the definition is violated at x0=0.

    Equicontinuity of maps valued in topological groups

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 maps

Because every topological vector space is a topological group, the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.

Characterization of equicontinuous linear maps

A family of maps of the form between two topological vector spaces is said to be if for every neighborhood of the origin in there exists some neighborhood of the origin in such that for all
If is a family of maps and is a set then let With notation, if and are sets then for all if and only if
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 the origin.
    • that is, for every neighborhood of the origin in there exists a neighborhood of the origin in such that .
  5. for every neighborhood of the origin in is a neighborhood of the origin 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 this list may be extended to include:

  1. the convex hull of is equicontinuous.
  2. the convex balanced hull of is equicontinuous.
while if and are locally convex then this list may be extended to include:

  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 this list may be extended to include:

  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 over the field with continuous dual space
A family of linear functionals on is said to be if for every neighborhood of the origin in there exists some neighborhood of the origin in such that for all
For any subset 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 the origin in
  5. the polar of is a neighborhood of the origin 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 this list may be extended to include:

  1. is a strongly bounded subset of
while if is a barreled space then this list may be extended to include:

  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; that is, 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 every 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.