Page 231 - DMTH401_REAL ANALYSIS

Unit 18: Equicontinuous

18.3 Equicontinuity and Uniform Convergence Notes

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 Ascoli’s theorem states that a subset of C(X) is
compact if and only if it is closed, pointwise bounded and equicontinuous. This is analogous to
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the Heine-Borel theorem, which states that subsets of  are compact if and only if they are
closed and bounded. Every bounded equicontinuous sequence in C(X) contains a subsequence
that converges uniformly to a continuous function on X.

In view of Ascoli’s 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 Ascoli’s 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  with g(0) = 1, and consider the equicontinuous
sequence of functions {f } on  defined by f (x) = g(x – n). Then, f converges pointwise to 0 but
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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
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of  . 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 homomorphic. One
can use, for instance, Cauchy’s estimate to show the equicontinuity (on a compact subset) and
conclude that the limit is homomorphism. Note that the equicontinuity is essential here. For
example, f (x) = arctan nx converges to a multiple of the discontinuous sign function.
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18.4 Equicontinuity Families of Linear Operators

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

Sup{||T|| : T  G } < 

that is, G is uniformly bounded in operator norm. Also, by linearity, G 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
G is equicontinuous if it is pointwise bounded; i.e., sup{||T(x)|| : T  G } <  for each x  E. The
result can be generalized to a case when F is locally convex and E is a barreled space.
Alaoglu’s theorem states that if E is a topological vector space, then every equicontinuous subset
of E* is weak-* relatively compact.

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