Download equicontinuous

equicontinuous∗
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2013-03-22 2:13:36
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Definition
Let X be a topological space, (Y, d) a metric space and C(X, Y ) the set of
continuous functions X → Y .
Let F be a subset of C(X, Y ). A function f ∈ F is continuous at a point x0
when given > 0 there is a neighbourhood U of x0 such that d(f (x), f (x0 )) <
for every x ∈ U . When the same neighbourhood U can be chosen for all
functions f ∈ F, the family F is said to be equicontinuous. More precisely:
Definition - Let F be a subset of C(X, Y ). The set of functions F is said
to be equicontinuous at x0 ∈ X if for every > 0 there is a neighbourhood
U of x0 such that for every x ∈ U and every f ∈ F we have
d(f (x), f (x0 )) < The set F is said to be equicontinuous if it is equicontinuous at every point
x ∈ X.
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Examples
• A finite set of functions in C(X, Y ) is always equicontinuous.
• When X is also a metric space, a family of functions in C(X, Y ) that share
the same Lipschitz constant is equicontinuous.
• The family of functions {fn }n∈N , where fn : R → R is given by fn (x) :=
arctan(nx) is not equicontinuous at 0.
∗ hEquicontinuousi created: h2013-03-2i by: hasteroidi version: h41376i Privacy setting:
h1i hDefinitioni h54E35i h54C35i
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Properties
• If a subset F ⊆ C(X, Y ) is totally bounded under the uniform metric,
then F is equicontinuous.
• Suppose X is compact. If a sequence of functions {fn } in C(X, Rk ) is
equibounded and equicontinuous, then the sequence {fn } has a uniformly
convergent subsequence. (Arzel’s theorem)
• Let {fn } be a sequence of functions in C(X, Y ). If {fn } is equicontinuous
and converges pointwise to a function f : X → Y , then f is continuous
and {fn } converges to f in the compact-open topology.
References
[1] J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
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