In the
mathematical field of
analysis, uniform convergence is a
mode of
convergence of functions stronger than
pointwise convergence. A
sequence of
functions converges uniformly to a limiting function
on a set
if, given any arbitrarily small positive number
, a number
can be found such that each of the functions
differs from
by no more than
''at every point''
''in''
. Described in an informal way, if
converges to
uniformly, then the rate at which
approaches
is "uniform" throughout its domain in the following sense: in order to guarantee that
falls within a certain distance
of
, we do not need to know the value of
in question — there can be found a single value of
''independent of
'', such that choosing
will ensure that
is within
of
''for all
''. In contrast, pointwise convergence of
to
merely guarantees that for any
given in advance, we can find
(
can depend on the value of ''
'') so that, ''for that particular'' ''
'',
falls within
of
whenever
.
The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by
Karl Weierstrass, is important because several properties of the functions
, such as
continuity,
Riemann integrability, and, with additional hypotheses,
differentiability, are transferred to the
limit
Limit or Limits may refer to:
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* "Limits", a 2019 ...
if the convergence is uniform, but not necessarily if the convergence is not uniform.
History
In 1821
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
published a proof that a convergent sum of continuous functions is always continuous, to which
Niels Henrik Abel in 1826 found purported counterexamples in the context of
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.
The term uniform convergence was probably first used by
Christoph Gudermann, in an 1838 paper on
elliptic functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series
is independent of the variables
and
While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.
Later Gudermann's pupil
Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term ''gleichmäßig konvergent'' (german: uniformly convergent) which he used in his 1841 paper ''Zur Theorie der Potenzreihen'', published in 1894. Independently, similar concepts were articulated by
Philipp Ludwig von Seidel and
George Gabriel Stokes.
G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis."
Under the influence of Weierstrass and
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
this concept and related questions were intensely studied at the end of the 19th century by
Hermann Hankel,
Paul du Bois-Reymond,
Ulisse Dini,
Cesare Arzelà and others.
Definition
We first define uniform convergence for
real-valued functions, although the concept is readily generalized to functions mapping to
metric spaces and, more generally,
uniform spaces (see
below
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).
Suppose
is a
set and
is a sequence of real-valued functions on it. We say the sequence
is uniformly convergent on
with limit
if for every
there exists a natural number
such that for all
and for all
:
The notation for uniform convergence of
to
is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity):
:
Frequently, no special symbol is used, and authors simply write
:
to indicate that convergence is uniform. (In contrast, the expression
on
without an adverb is taken to mean
pointwise convergence on
: for all
,
as
.)
Since
is a
complete metric space, the
Cauchy criterion
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'An ...
can be used to give an equivalent alternative formulation for uniform convergence:
converges uniformly on
(in the previous sense) if and only if for every
, there exists a natural number
such that
:
.
In yet another equivalent formulation, if we define
:
then
converges to
uniformly if and only if
as
. Thus, we can characterize uniform convergence of
on
as (simple) convergence of
in the
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
with respect to the ''
uniform metric'' (also called the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
metric), defined by
:
Symbolically,
:
.
The sequence
is said to be locally uniformly convergent with limit
if
is a
metric space and for every
, there exists an
such that
converges uniformly on
It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.
Notes
Intuitively, a sequence of functions
converges uniformly to
if, given an arbitrarily small
, we can find an
so that the functions
with
all fall within a "tube" of width
centered around
(i.e., between
and
) for the ''entire domain'' of the function.
Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all
" in front of "there exists a natural number
" results in a definition of
pointwise convergence of the sequence. To make this difference explicit, in the case of uniform convergence,
can only depend on
, and the choice of
has to work for all
, for a specific value of
that is given. In contrast, in the case of pointwise convergence,
may depend on both
and
, and the choice of
only has to work for the specific values of
and
that are given. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates.
Generalizations
One may straightforwardly extend the concept to functions ''E'' → ''M'', where (''M'', ''d'') is a
metric space, by replacing
with
.
The most general setting is the uniform convergence of
nets of functions ''E'' → ''X'', where ''X'' is a
uniform space. We say that the net
''converges uniformly'' with limit ''f'' : ''E'' → ''X'' if and only if for every
entourage ''V'' in ''X'', there exists an
, such that for every ''x'' in ''E'' and every
,
is in ''V''. In this situation, uniform limit of continuous functions remains continuous.
Definition in a hyperreal setting
Uniform convergence admits a simplified definition in a
hyperreal setting. Thus, a sequence
converges to ''f'' uniformly if for all ''x'' in the domain of
and all infinite ''n'',
is infinitely close to
(see
microcontinuity for a similar definition of uniform continuity).
Examples
For
, a basic example of uniform convergence can be illustrated as follows: the sequence
converges uniformly, while
does not. Specifically, assume
. Each function
is less than or equal to
when
, regardless of the value of
. On the other hand,
is only less than or equal to
at ever increasing values of
when values of
are selected closer and closer to 1 (explained more in depth further below).
Given a topological space ''X'', we can equip the space of bounded function, bounded real number, real or complex number, complex-valued functions over ''X'' with the
uniform norm topology, with the uniform metric defined by
:
Then uniform convergence simply means
convergence in the
uniform norm topology:
:
.
The sequence of functions
:
is a classic example of a sequence of functions that converges to a function
pointwise but not uniformly. To show this, we first observe that the pointwise limit of
as
is the function
, given by
:
''Pointwise convergence:'' Convergence is trivial for
and
, since
and
, for all
. For
and given
, we can ensure that
whenever
by choosing
(here the upper square brackets indicate rounding up, see Floor and ceiling functions, ceiling function). Hence,
pointwise for all