Page 230 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 18.2 Families of Equicontinuous

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 x  X if for every  > 0, there exists a  > 0 such that
0
d(f(x ), f(x)) <  for all f  F and all x such that d(x , x) < . The family is equicontinuous if it is
0 0
equicontinuous at each point of X.
The family F is uniformly equicontinuous if for every  > 0, there exists a  > 0 such that d(f(x ),
1
f(x )) <  for all f  F and all x , x  X such that d(x , x ) < .
2 1 2 1 2
For comparison, the statement all functions f in F are continuous’ means that for every  > 0,
every ƒ  F, and every x  X, there exists a  > 0 such that d(f(x ), f(x)) <  for all x  X such that
0 0
d(x , x) < . So, for continuity,  may depend on , x and f; for equicontinuity,  must be
0 0
independent of f; and for uniform equicontinuity,  must be independent of both f and x .
0
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  > 0, x has a neighbourhood U such that
x
d (f(y), f(x)) < 
Y
for all y  U and f  F. This definition usually appears in the context of topological vector
x
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.
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.

Example:
 A set of functions with the same 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.

Properties of Equicontinuous

 If a subset   C(X, Y) is totally bounded under the uniform metric, and then  is
equicontinuous.
 Suppose X is compact. If a sequence of functions {f } in C(X k) is equibounded and
n
equicontinuous, then the sequence {f } has a uniformly convergent subsequence. (Arzelà’s
n
theorem)
 Let {f } be a sequence of functions in C(X, Y). If {f } is equicontinuous and converges
n n
pointwise to a function f : X  Y, then f is continuous and {f } converges to f in the compact-
n
open topology.

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