In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s
from a set ''S'' to a metric space ''M'' is said to be uniformly Cauchy if:
* For all
, there exists
such that for all
:
whenever
.
Another way of saying this is that
as
, where the uniform distance
between two functions is defined by
:
Convergence criteria
A sequence of functions from ''S'' to ''M'' is pointwise Cauchy if, for each ''x'' ∈ ''S'', the sequence is a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
in ''M''. This is a weaker condition than being uniformly Cauchy.
In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space ''M'' is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, then any pointwise Cauchy sequence converges pointwise to a function from ''S'' to ''M''. Similarly, any uniformly Cauchy sequence will tend
uniformly to such a function.
The uniform Cauchy property is frequently used when the ''S'' is not just a set, but a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, and ''M'' is a complete metric space. The following theorem holds:
* Let ''S'' be a topological space and ''M'' a complete metric space. Then any uniformly Cauchy sequence of
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s ''f''
n : ''S'' → ''M'' tends
uniformly to a unique continuous function ''f'' : ''S'' → ''M''.
Generalization to uniform spaces
A
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s
from a set ''S'' to a
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
''U'' is said to be uniformly Cauchy if:
* For all
and for any entourage
, there exists
such that
whenever
.
See also
*
Modes of convergence (annotated index)
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of converg ...
Functional analysis
Convergence (mathematics)
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