Page 232 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 18.5 Equicontinuity in Topological Spaces

The most general scenario in which equicontinuity can be defined is for topological spaces
whereas uniform equicontinuity requires the filter of neighbourhoods of one point to be somehow
comparable with the filter of neighbourhood of another point. The latter is most generally done
via a uniform structure, giving a uniform space. Appropriate definitions in these cases are as
follows:
A set A of functions continuous between two topological spaces X and Y is topologically
equicontinuous at the points x  X and y  Y if for any open set O about y, there are neighbourhoods
U of x and V of y such that for every f  A, if the intersection of f[U] and V is non-empty, f(U)  O.
One says A is said to be topologically equicontinuous at x  X if it is topologically equicontinuous
at x and y for each y  Y. Finally, A is equicontinuous if it is equicontinuous at x for all points
x  X.

A set A of continuous functions between two uniform spaces X and Y is uniformly equicontinuous
if for every element W of the uniformity on Y, the set
{(u, v)  X  X : for all f  A. (f(u), f(v))  W }
is a member of the uniformity on X
A weaker concept is that of even continuity:

A set A of continuous functions between two topological spaces X and Y is said to be evenly
continuous at x  X and y  Y if given any open set O containing y there are neighbourhoods U
of x and V of y such that f[U]  O whenever f(x)  V. It is evenly continuous at x if it is evenly
continuous at x and y for every y  Y, and evenly continuous if it is evenly continuous at x for
every x  X.
For metric spaces, there are standard topologies and uniform structures derived from the matrices,
and then these general definitions are equivalent to the metric-space definitions.

18.6 Stochastic Equicontinuity

Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of
functions of random variables, and their convergence.
Let {H () : n  1} be a family of random functions defined from, where  where  is any
n
normed metric space. Here {H ()} might represent a sequence of estimators applied to datasets
n
of size n, given that the data arises from a population for which the parameter indexing the
statistical model for the data is . The randomness of the functions arises from the data generating
process under which a set of observed data is considered to be a realisation of a probabilistic or
statistical model. However, in {H ()},  relates to the model currently being postulated or fitted
n
rather than to an underlying model which is supposed to represent the mechanism generating
the data. Then {H } is stochastically equicontinuous if, for every  > 0, there is a  > 0 such that:
n
æ ö

-

lim Pr sup sup H ( ') H ( ) >  < 
÷
ç
n
n
n è   ' B( , ) ø



Here B(, ) represents a ball in the parameter space, centered at  and whose radius depends on.
Self Assessment
Fill in the blanks:
1. The ……………………..states that a pointwise bounded family of continuous linear
operators between Banach spaces is equicontinuous.
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