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_n on either metric space or locally compact space^[1] is continuous. If, in addition, f_n 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.

Definition

Let X and Y be two metric spaces, and F a family of functions from X to Y.

The family F is equicontinuous at a point x₀ ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x₀), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x₀, x) < δ. The family is pointwise equicontinuous if it is equicontinuous at each point of X.^[2]

The family F is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that d(ƒ(x₁), ƒ(x₂)) < ε for all ƒ ∈ F and all x₁, x₂ ∈ X such that d(x₁, x₂) < δ.^[3]

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(x₀), ƒ(x)) < ε for all x ∈ X such that d(x₀, x) < δ.

For continuity, δ may depend on ε, x₀ and ƒ.
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 solely depend 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 U_x such that

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

for all y ∈ U_x 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

A set of functions with a common Lipschitz constant is (uniformly) 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.^[4]^[5]

Counter Examples

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

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 Rⁿ 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).^[6] 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 {ƒ_n} on R defined by ƒ_n(x) = g(x − n). Then, ƒ_n 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 (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_n(x) = arctan n x converges to a multiple of the discontinuous sign function.

Generalizations

Equicontinuity of families of linear operators

Let E, F be Banach spaces, and Γ be a family of continuous linear operators from E into F. Then Γ is equicontinuous if and only if

\sup\{\|T\|:T\in \Gamma \}<\infty

that is, Γ is uniformly bounded in operator norm. Also, by linearity, Γ is uniformly equicontinuous if and only if it is equicontinuous at 0.

The uniform boundedness principle (also known as the Banach–Steinhaus theorem) states that Γ is equicontinuous if it is pointwise bounded; i.e., sup {‖T(x)‖: T ∈ Γ} < ∞ for each x ∈ E. The result can be generalized to a case when F is locally convex and E is a barreled space.^[7]

Alaoglu's theorem states that if E is a topological vector space, then every equicontinuous subset of E* is weak-* relatively compact.^[8]

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:

A set A of functions continuous between two topological spaces X and Y is topologically equicontinuous at the points x ∈ X and y ∈ Y if for any open set O about y, there are neighborhoods U of x and V of y such that for every f ∈ A, if the intersection of f[U] and V is nonempty, f(U) ⊆ O. One says A is said to be topologically equicontinuous at x ∈ X if it is topologically equicontinuous at x and y for each y ∈ Y. Finally, A is equicontinuous if it is equicontinuous at x for all points x ∈ X.

A set A of continuous functions between two uniform spaces X and Y is uniformly equicontinuous if for every element W of the uniformity on Y, the set

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

is a member of the uniformity on X

A weaker concept is that of even continuity:

A set A of continuous functions between two topological spaces X and Y is said to be evenly continuous at x ∈ X and y ∈ Y if given any open set O containing y there are neighborhoods U of x and V of y such that f[U] ⊆ O whenever f(x) ∈ V. It is evenly continuous at x if it is evenly continuous at x and y for every y ∈ Y, and evenly continuous if it is evenly continuous at x for every x ∈ X.

For metric spaces, there are standard topologies and uniform structures derived from the metrics, and then these general definitions are equivalent to the metric-space definitions.^{[citation needed]}

Stochastic equicontinuity

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

Notes

^ More generally, on any compactly generated space; e.g., a first-countable space.
^ 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
^ Proof: Suppose f_j is an equicontinuous sequence of continuous functions on a dense subset D of X. Let ε > 0 be given. By equicontinuity, for each z ∈ D, there exists a neighborhood U_z of z such that
$|f_{j}(x)-f_{j}(z)|<\epsilon /3\,$
for all j and x ∈ U_z. By denseness and compactness, we can find a finite subset D′ ⊂ D such that X is the union of U_z over z ∈ D′. Since f_j converges pointwise on D′, there exists N > 0 such that
$|f_{j}(z)-f_{k}(z)|<\epsilon /3\,$
whenever z ∈ D′ and j, k > N. It follows that
$\sup _{X}|f_{j}-f_{k}|<\epsilon$
for all j, k > N. In fact, if x ∈ X, then x ∈ U_z for some z ∈ 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_j is Cauchy in C(X) and thus converges by completeness.
^ Schaefer, Theorem 4.2
^ Schaefer, Corollary 4.3

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.
Rudin, Walter (1987), Real and Complex Analysis (3rd ed.), New York: McGraw-Hill.
Schaefer, Helmuth H. (1966), Topological vector spaces, New York: The Macmillan Company

[1] More generally, on any compactly generated space; e.g., a first-countable space.

[RS29-2] Reed & Simon (1980), p. 29; Rudin (1987), p. 245

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

[4] 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

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

[6] Proof: Suppose f_j is an equicontinuous sequence of continuous functions on a dense subset D of X. Let ε > 0 be given. By equicontinuity, for each z ∈ D, there exists a neighborhood U_z of z such that
$|f_{j}(x)-f_{j}(z)|<\epsilon /3\,$
for all j and x ∈ U_z. By denseness and compactness, we can find a finite subset D′ ⊂ D such that X is the union of U_z over z ∈ D′. Since f_j converges pointwise on D′, there exists N > 0 such that
$|f_{j}(z)-f_{k}(z)|<\epsilon /3\,$
whenever z ∈ D′ and j, k > N. It follows that
$\sup _{X}|f_{j}-f_{k}|<\epsilon$
for all j, k > N. In fact, if x ∈ X, then x ∈ U_z for some z ∈ 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_j is Cauchy in C(X) and thus converges by completeness.

[7] Schaefer, Theorem 4.2

[8] Schaefer, Corollary 4.3

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