In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a family of functions is equicontinuous if all the functions are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
and they have equal variation over a given
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
, in a precise sense described herein.
In particular, the concept applies to
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
families, and thus ''sequences'' of functions.
Equicontinuity appears in the formulation of
Ascoli's theorem, which states that a subset of ''C''(''X''), the space of
continuous functions on a compact Hausdorff space In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X with values in the real or complex numbers. This space, denoted by \mathcal(X), is a vec ...
''X'', is compact if and only if it is closed, pointwise bounded and equicontinuous.
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).
In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions ''f
n'' on either metric space or
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
is continuous. If, in addition, ''f
n'' are
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
, then the limit is also holomorphic.
The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.
Equicontinuity between metric spaces
Let ''X'' and ''Y'' be two
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s, and ''F'' a family of functions from ''X'' to ''Y''. We shall denote by ''d'' the respective metrics of these spaces.
The family ''F'' is equicontinuous at a point ''x''
0 ∈ ''X'' if for every ε > 0, there exists a δ > 0 such that ''d''(''ƒ''(''x''
0), ''ƒ''(''x'')) < ε for all ''ƒ'' ∈ ''F'' and all ''x'' such that ''d''(''x''
0, ''x'') < δ.
The family is pointwise equicontinuous if it is equicontinuous at each point of ''X''.
[, p. 29; , p. 245]
The family ''F'' is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that ''d''(''ƒ''(''x''
1), ''ƒ''(''x''
2)) < ε for all ''ƒ'' ∈ ''F'' and all ''x''
1, ''x''
2 ∈ ''X'' such that ''d''(''x''
1, ''x''
2) < δ.
For comparison, the statement 'all functions ''ƒ'' in ''F'' are continuous' means that for every ε > 0, every ''ƒ'' ∈ ''F'', and every ''x''
0 ∈ ''X'', there exists a δ > 0 such that ''d''(''ƒ''(''x''
0), ''ƒ''(''x'')) < ε for all ''x'' ∈ ''X'' such that ''d''(''x''
0, ''x'') < δ.
* For ''
continuity'', δ may depend on ε, ''ƒ'', and ''x''
0.
* For ''
uniform continuity
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
'', δ may depend on ε and ''ƒ''.
* For ''pointwise equicontinuity'', δ may depend on ε and ''x''
0.
* For ''uniform equicontinuity'', δ may depend only on ε.
More generally, when ''X'' is a topological space, a set ''F'' of functions from ''X'' to ''Y'' is said to be equicontinuous at ''x'' if for every ε > 0, ''x'' has a neighborhood ''U
x'' such that
:
for all and ''ƒ'' ∈ ''F''. This definition usually appears in the context of
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s.
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.
Used on its own, the term "equicontinuity" may refer to either the pointwise or uniform notion, depending on the context. On a compact space, these notions coincide.
Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous. The closure of an equicontinuous set is again equicontinuous.
Every member of a uniformly equicontinuous set of functions is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
, and every finite set of uniformly continuous functions is uniformly equicontinuous.
Examples
*A set of functions with a common
Lipschitz constant
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...
is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant.
*
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
gives a sufficient condition for a set of continuous linear operators to be equicontinuous.
*A family of iterates of an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
is equicontinuous on the
Fatou set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wit ...
.
Counterexamples
*The sequence of functions f
n(x) = arctan(nx), is not equicontinuous because the definition is violated at x
0=0
Equicontinuity of maps valued in topological groups
Suppose that is a topological space and is an additive
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
(i.e. a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
endowed with a topology making its operations continuous).
Topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s are prominent examples of topological groups and every topological group has an associated canonical
uniformity.
:Definition: A family of maps from into is said to be equicontinuous at if for every neighborhood of in , there exists some neighborhood of in such that for every . We say that is equicontinuous if it is equicontinuous at every point of .
Note that if is equicontinuous at a point then every map in is continuous at the point.
Clearly, every finite set of continuous maps from into is equicontinuous.
Equicontinuous linear maps
Because every
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change.
Characterization of equicontinuous linear maps
A family
of maps of the form
between two topological vector spaces is said to be
if for every neighborhood
of the origin in
there exists some neighborhood
of the origin in
such that
for all
If
is a family of maps and
is a set then let
With notation, if
and
are sets then
for all
if and only if
Let
and
be
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs) and
be a family of linear operators from
into
Then the following are equivalent:
- is equicontinuous;
- is equicontinuous at every point of
- is equicontinuous at some point of
- is equicontinuous at the origin.
* that is, for every neighborhood of the origin in there exists a neighborhood of the origin in such that (or equivalently, for every ).
- for every neighborhood of the origin in is a neighborhood of the originin
- the closure of in is equicontinuous.
* denotes endowed with the topology of point-wise convergence.
- the
balanced hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of is equicontinuous.
while if
is
locally convex then this list may be extended to include:
- the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of is equicontinuous.
- the
convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...
of is equicontinuous.
while if
and
are
locally convex then this list may be extended to include:
- for every continuous
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
on there exists a continuous seminorm on such that for all
* Here, means that for all
while if
is
barreled and
is locally convex then this list may be extended to include:
- is bounded in ;
- is bounded in
* denotes endowed with the topology of bounded convergence (that is, uniform convergence on bounded subsets of
while if
and
are
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s then this list may be extended to include:
- (that is, is uniformly bounded in the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
).
Characterization of equicontinuous linear functionals
Let
be a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) over the field
with
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
A family
of linear functionals on
is said to be
if for every neighborhood
of the origin in
there exists some neighborhood
of the origin in
such that
for all
For any subset
the following are equivalent:
- is equicontinuous.
- is equicontinuous at the origin.
- is equicontinuous at some point of
- is contained in the
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
* Polar climate, the c ...
of some neighborhood of the origin in
- the (pre)polar of is a neighborhood of the origin in
- the weak* closure of in is equicontinuous.
- the
balanced hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of is equicontinuous.
- the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of is equicontinuous.
- the
convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...
of is equicontinuous.
while if
is
normed then this list may be extended to include:
- is a strongly bounded subset of
while if
is a
barreled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a ...
then this list may be extended to include:
- is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (since ...
in the weak* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on
- is weak* bounded (that is, is bounded in ).
- is bounded in the topology of bounded convergence (that is, is bounded in ).
Properties of equicontinuous linear maps
The
uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
(also known as the Banach–Steinhaus theorem) states that a set
of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is,
for each
The result can be generalized to a case when
is locally convex and
is a
barreled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a ...
.
Properties of equicontinuous linear functionals
Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of
is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.
If
is any locally convex TVS, then the family of all
barrel
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
s in
and the family of all subsets of
that are convex, balanced, closed, and bounded in
correspond to each other by polarity (with respect to
).
It follows that a locally convex TVS
is barreled if and only if every bounded subset of
is equicontinuous.
Equicontinuity and uniform convergence
Let ''X'' be a compact Hausdorff space, and equip ''C''(''X'') with the
uniform norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
, thus making ''C''(''X'') a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, hence a metric space. Then
Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inter ...
states that a subset of ''C''(''X'') is compact if and only if it is closed, uniformly bounded and equicontinuous.
This is analogous to the
Heine–Borel theorem, which states that subsets of R
''n'' are compact if and only if they are closed and bounded.
As a corollary, every uniformly bounded equicontinuous sequence in ''C''(''X'') contains a subsequence that converges uniformly to a continuous function on ''X''.
In view of Arzelà–Ascoli theorem, a sequence in ''C''(''X'') converges uniformly if and only if it is equicontinuous and converges pointwise. The hypothesis of the statement can be weakened a bit: a sequence in ''C''(''X'') converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on ''X'' (not assumed continuous).
This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous. (See below for an example.)
In the above, the hypothesis of compactness of ''X'' cannot be relaxed.
To see that, consider a compactly supported continuous function ''g'' on R with ''g''(0) = 1, and consider the equicontinuous sequence of functions on R defined by ''ƒ''
''n''(''x'') = . Then, ''ƒ''
''n'' converges pointwise to 0 but does not converge uniformly to 0.
This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subset ''G'' of R
''n''. As noted above, it actually converges uniformly on a compact subset of ''G'' if it is equicontinuous on the compact set. In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset of ''G''; thus, continuous on ''G''. A similar argument can be made when the functions are holomorphic. One can use, for instance,
Cauchy's estimate
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the ...
to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example, ''ƒ''
''n''(''x'') = converges to a multiple of the discontinuous
sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
.
Generalizations
Equicontinuity in topological spaces
The most general scenario in which equicontinuity can be defined is for
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 points ...
s whereas ''uniform'' equicontinuity requires the
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
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
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 unifor ...
, giving 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 ...
. 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 neighborhoods ''U'' of ''x'' and ''V'' of ''y'' such that for every ''f'' ∈ ''A'', if the intersection of ''f''
'U''and ''V'' is nonempty, ''f''
'U''⊆ ''O''. Then ''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
::
:is a member of the uniformity on ''X''
;Introduction to uniform spaces
We now briefly describe the basic idea underlying uniformities.
The uniformity is a non-empty collection of subsets of where, among many other properties, every , contains the diagonal of (i.e. ).
Every element of is called an entourage.
Uniformities generalize the idea (taken from
metric spaces
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
) of points that are "-close" (for ), meaning that their distance is < .
To clarify this, suppose that is a metric space (so the diagonal of is the set )
For any , let
:
denote the set of all pairs of points that are -close.
Note that if we were to "forget" that existed then, for any , we would still be able to determine whether or not two points of are -close by using only the sets .
In this way, the sets encapsulate all the information necessary to define things such as
uniform continuity
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
and
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
with''out'' needing any metric.
Axiomatizing the most basic properties of these sets leads to the definition of a
uniformity.
Indeed, the sets generate the uniformity that is canonically associated with the metric space .
The benefit of this generalization is that we may now extend some important definitions that make sense for metric spaces (e.g.
completeness) to a broader category of topological spaces.
In particular, to
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s and
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s.
;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 neighborhoods ''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''.
Stochastic equicontinuity
Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of
random variables
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, and their
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
.
See also
*
*
*
*
*
*
*
* - an analogue of a continuous function in discrete spaces.
*
*
*
*
*
Notes
References
*
* .
*
* .
*
*
*
Theory of continuous functions
Mathematical analysis