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.
In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space[1] is continuous.
The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.
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 y ∈ Ux 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.
is a barreled space then this list may be extended to include: The uniform boundedness principle (also known as the Banach–Steinhaus theorem) states that a set
of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is,
[12] Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of
is a compact metrizable space (under the subspace topology).
Then Arzelà–Ascoli theorem states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous.
[15] 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.
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).
Suppose fj is an equicontinuous sequence of continuous functions on a dense subset D of X.
By denseness and compactness, we can find a finite subset D′ ⊂ D such that X is the union of Uz over z ∈ D′.
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.
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.
One can use, for instance, Cauchy's estimate to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic.
For example, ƒn(x) = arctan n x converges to a multiple of the discontinuous sign function.
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.
Appropriate definitions in these cases are as follows: We now briefly describe the basic idea underlying uniformities.
Uniformities generalize the idea (taken from metric spaces) of points that are "r-close" (for r > 0), meaning that their distance is < r. To clarify this, suppose that (Y, d) is a metric space (so the diagonal of Y is the set {(y, z) ∈ Y × Y : d(y, z) = 0}) For any r > 0, let denote the set of all pairs of points that are r-close.
Note that if we were to "forget" that d existed then, for any r > 0, we would still be able to determine whether or not two points of Y are r-close by using only the sets Ur.
Axiomatizing the most basic properties of these sets leads to the definition of a uniformity.
Indeed, the sets Ur generate 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.