Page 233 - DMTH401_REAL ANALYSIS

Unit 18: Equicontinuous

2. The family F is ……………………x  X if for every  > 0, there exists a  > 0 such that Notes
0
d(f(x ), f(x)) <  for all ƒ  F and all x such that d(x , x) < . The family is equicontinuous if
0 0
it is equicontinuous at each point of X.
3. 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 …………………………..
n
4. The uniform boundedness principle is also known as …………………….. 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.
5. …………….. is a version of equicontinuity used in the context of sequences of functions of
random variables, and their convergence.

18.7 Summary

 In particular, the limit of an equicontinuous pointwise convergent sequence of continuous
functions f on either metric space or locally compact space is continuous. If, in addition, f
n 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.

 The family F is equicontinuous at a point x  X if for every  > 0, there exists a  > 0 such
0
that d(f(x ), f(x)) <  for all f  F and all x such that d(x , x) < . The family is equicontinuous
0 0
if it is 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 ), f(x )) <  for all f  F and all x , x  X such that d(x , x ) < .
1 2 1 2 1 2
 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 fn be a sequence of functions in C(X, Y). If {f } is equicontinuous and converges pointwise
n
to a function f : X  Y, then f is continuous and {f } converges to f in the compact-open
n
topology.
 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.

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

18.8 Keywords

Stochastic Equicontinuity: Stochastic equicontinuity is a version of equicontinuity used in the
context of sequences of functions of random variables, and their convergence
Uniformly Equicontinuous: The family F is uniformly equicontinuous if for every  > 0, there
exists a  > 0 such that d(f(x ), f(x )) <  for all f  F and all x , x  X such that d(x , x ) < .
1 2 1 2 1 2

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