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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 ...
, pointwise convergence is one of various senses in which 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 calle ...
of functions can
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
to a particular function. It is weaker than
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 ...
, to which it is often compared.


Definition

Suppose that X is a set and Y is 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 points ...
, such as the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
or a
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 ...
, for example. A
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
or
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 calle ...
of functions \left(f_n\right) all having the same domain X and
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
Y is said to converge pointwise to a given function f : X \to Y often written as \lim_ f_n = f\ \mbox if (and only if) \lim_ f_n(x) = f(x) \text x \text f. The function f is said to be the pointwise limit function of the \left(f_n\right). Sometimes, authors use the term bounded pointwise convergence when there is a constant C such that \forall n,x,\;, f_n(x), .


Properties

This concept is often contrasted with
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 ...
. To say that \lim_ f_n = f\ \mbox means that \lim_\,\sup\=0, where A is the common domain of f and f_n, and \sup stands for 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 ...
. That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent. For example, if f_n : [0,1) \to [0,1) is a sequence of functions defined by f_n(x) = x^n, then \lim_ f_n(x) = 0 pointwise on the interval [0, 1), but not uniformly. The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f(x) = \lim_ \cos(\pi x)^ takes the value 1 when x is an integer and 0 when x is not an integer, and so is discontinuous at every integer. The values of the functions f_n need not be real numbers, but may be in any
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 ...
, in order that the concept of pointwise convergence make sense. Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in
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, more generally, in uniform spaces.


Topology

Let Y^X denote the set of all functions from some given set X into some
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 ...
Y. As described in the article on
characterizations of the category of topological spaces In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a t ...
, if certain conditions are met then it is possible to define a unique topology on a set in terms of what
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
s do and do not
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
. The definition of pointwise convergence meets these conditions and so it induces a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, called the , on the set Y^X of all functions of the form X \to Y. A net in Y^X converges in this topology if and only if it converges pointwise. The topology of pointwise convergence is the same as convergence in the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
on the space Y^X, where X is the domain and Y is the codomain. Explicitly, if \mathcal \subseteq Y^X is a set of functions from some set X into some topological space Y then the topology of pointwise convergence on \mathcal is equal to the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
that it inherits from the
product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
\prod_ Y when \mathcal is identified as a subset of this Cartesian product via the canonical inclusion map \mathcal \to \prod_ Y defined by f \mapsto (f(x))_. If the codomain Y is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, then by
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
, the space Y^X is also compact.


Almost everywhere convergence

In
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, one talks about ''almost everywhere convergence'' of a sequence of
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s defined on a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
. That means pointwise convergence
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, that is, on a subset of the domain whose complement has measure zero.
Egorov's theorem In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, a ...
states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set. Almost everywhere pointwise convergence on the space of functions on a measure space does not define the structure of a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
on the space of measurable functions on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
(although it is a convergence structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the same
subsequential limit In mathematics, a subsequential limit of a sequence is the Limit of a sequence, limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In First-countable space, first-countable spaces, the two concepts coincid ...
, the sequence itself must converge to that limit. But consider the sequence of so-called "galloping rectangles" functions, which are defined using the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
: let N = \operatorname\left(\log_2 n\right) and k = n
mod Mod, MOD or mods may refer to: Places * Modesto City–County Airport, Stanislaus County, California, US Arts, entertainment, and media Music * Mods (band), a Norwegian rock band * M.O.D. (Method of Destruction), a band from New York City, US ...
2^X, and let f_n(x) = \begin 1, & \frac \leq x \leq \frac \\ 0, & \text. \end Then any subsequence of the sequence \left(f_n\right)_n has a sub-subsequence which itself converges almost everywhere to zero, for example, the subsequence of functions which do not vanish at x = 0. But at no point does the original sequence converge pointwise to zero. Hence, unlike
convergence in measure Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability. Definitions Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be measurable functions on a measure space (X, \ ...
and L^p convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.


See also

* * * * * * * *


References

{{DEFAULTSORT:Pointwise Convergence Convergence (mathematics) Measure theory Topological spaces Topology of function spaces hu:Függvénysorozatok konvergenciája#Pontonkénti konvergencia