This family of maps { " P " " N " } is equicontinuous and tends to the identity on the dense subset consisting of trigonometric polynomials.
12.
Suppose that " f n " & prime; are uniformly equicontinuous and uniformly bounded, and that the sequence is pointwise bounded ( or just bounded at a single point ).
13.
Alaoglu's theorem states that if " E " is a topological vector space, then every equicontinuous subset of " E * " is weak-* relatively compact.
14.
In the non-i . i . d . case the uniform convergence in probability can be checked by showing that the sequence \ scriptstyle \ hat \ ell ( \ theta \ mid x ) is stochastically equicontinuous.
15.
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.
16.
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.
17.
As a corollary, a sequence in " C " ( " X " ) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function ( not necessarily continuous a-priori ).
18.
Suppose that X, Y, and Z are locally convex spaces and let \ mathcal { G }'and \ mathcal { H }'be the collections of equicontinuous subsets of X ^ * and Y ^ *, respectively.
19.
Then a subset "'F "'of " C " ( " X " ) is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and pointwise bounded.
20.
Given a barrelled space " X " and a locally convex space " Y ", then any family of pointwise bounded continuous linear mappings from " X " to " Y " is equicontinuous ( even uniformly equicontinuous ).
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