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Notion of convergence in mathematics
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than
Pointwise_convergence
Mode of convergence of a function sequence
mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n )
Uniform_convergence
Theorem in measure theory
is almost everywhere pointwise convergent to a function then the sequence converges in L 1 {\displaystyle L_{1}} to its pointwise limit, and in particular
Dominated_convergence_theorem
Mathematical problem in classical harmonic analysis
met. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability
Convergence_of_Fourier_series
Property of a sequence or series
define pointwise Cauchy convergence, uniform convergence, and uniform Cauchy convergence of the sequence. Pointwise convergence implies pointwise Cauchy
Modes_of_convergence
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The
Convergence of random variables
Convergence_of_random_variables
Applying operations to functions in terms of values for each input "point"
if and only if k ≤ idA. An example of an infinitary pointwise relation is pointwise convergence of functions—a sequence of functions ( f n ) n = 1 ∞
Pointwise
) {\displaystyle L(X;Y)} or the topology of pointwise convergence or the topology of simple convergence and L ( X ; Y ) {\displaystyle L(X;Y)} with this
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
Result in probability theory
convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. This theorem is the basis for
Lévy's_continuity_theorem
Theorem stating that pointwise boundedness implies uniform boundedness
of bounded operators ( T n ) {\displaystyle \left(T_{n}\right)} converges pointwise, that is, the limit of ( T n ( x ) ) {\displaystyle \left(T_{n}(x)\right)}
Uniform_boundedness_principle
Lemma in measure theory
Using the definition of X, its representation as pointwise limit of the Yk, the monotone convergence theorem for conditional expectations, the last inequality
Fatou's_lemma
Mathematical concept
kernel K. For pointwise convergence there is simple argument due to Mateu & Verdera (2006) showing that the truncated integrals converge to Tf precisely
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
1966 result in mathematical analysis
result in mathematical analysis establishing the (Lebesgue) pointwise almost everywhere convergence of Fourier series of L2 functions, proved by Lennart Carleson
Carleson's_theorem
Theorem in functional analysis
},X\right).} The weak-* topology is also called the topology of pointwise convergence because given a map f {\displaystyle f} and a net of maps f ∙ =
Banach–Alaoglu_theorem
Value approached by a mathematical object
continuous functions which has a discontinuous pointwise limit. Another notion of convergence is uniform convergence. The uniform distance between two functions
Limit_(mathematics)
to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence. The convergence was defined by Robert Wijsman. The same definition
Wijsman_convergence
Mathematical theorem in real analysis
continuous as well. This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒn : [0, 1] → R be the sequence of
Uniform_limit_theorem
convergence -- In pointwise convergence, some (open) regions can converge arbitrarily slowly. With uniform convergence, there is a fixed convergence rate
Convergence_proof_techniques
Value to which an infinite sequence tends
Limit of a sequence of sets Limit of a net Pointwise convergence Uniform convergence Modes of convergence Courant (1961), p. 29. Weisstein, Eric W. "Convergent
Limit_of_a_sequence
Theorems on the convergence of bounded monotonic sequences
analysis, the monotone convergence theorem is any of a number of related theorems proving, under certain conditions, the convergence of monotonic sequences
Monotone_convergence_theorem
Decomposition of periodic functions
functions or distributions, in which case pointwise convergence often fails, and convergence in norm or weak convergence is usually studied. Four partial sums
Fourier_series
Mathematical concept
Weak-* convergence is sometimes called the simple convergence or the pointwise convergence. Indeed, it coincides with the pointwise convergence of linear
Weak_topology
Application of set theory concept
filter converges—called its Hausdorff completion. The underlying set is the set of minimal Cauchy filters. The notions of pointwise convergence and uniform
Filters_in_topology
Sum of an (infinite) geometric progression
subtleties into the questions of convergence, such as the distinctions between uniform convergence and pointwise convergence in series of functions, and can
Geometric_series
Theorem concerning uniform convergence
mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named
Egorov's_theorem
Topology on Cartesian products of topological spaces
topology of pointwise convergence because a sequence (or more generally, a net) in ∏ i ∈ I X i {\textstyle \prod _{i\in I}X_{i}} converges if and only
Product_topology
Indicator function of positive numbers
limits hold pointwise and in the sense of distributions. In general, however, pointwise convergence need not imply distributional convergence, and vice
Heaviside_step_function
Mathematics of real numbers and real functions
modes of convergence. A sequence of functions converges pointwise if it converges at every point, but, roughly speaking, the rate of convergence may vary
Real_analysis
Test for infinite series of monotonous terms for convergence
mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin
Integral_test_for_convergence
Sufficient criterion for uniform convergence
continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. The theorem
Dini's_theorem
Finite or infinite ordered list of elements
norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called an
Sequence
Two-volume set of books by Antoni Zygmund
developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square-integrable functions.[citation needed] Zygmund, Antoni
Trigonometric_Series
Measure in mathematical analysis
( U , R n ) {\displaystyle L^{\infty }(U,\mathbb {R} ^{n})} and converges pointwise almost everywhere in U {\displaystyle U} to a function f {\displaystyle
Young_measure
African-American mathematician
research contributions include pointwise convergence of averages along cubes, being “the first complete pointwise convergence result obtained in the theory
Idris_Assani
Topics referred to by the same term
Look up convergence, converges, or converging in Wiktionary, the free dictionary. Convergence may refer to: Convergence (book series), edited by Ruth
Convergence
mathematical analysis, epi-convergence is a type of convergence for real-valued and extended real-valued functions. Epi-convergence is important because it
Epi-convergence
Speed of convergence of a mathematical sequence
particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations
Rate_of_convergence
Romanian-American mathematician (1935–2025)
the following properties: (I) H is compact (for the topology of pointwise convergence); (II) H is convex; (III) H satisfies the "separation property"
Alexandra_Bellow
Area of mathematical analysis
Hardy–Littlewood maximal function. Maximal functions are used to control pointwise convergence, differentiation of integrals, and boundary limits of harmonic or
Harmonic_analysis
Mathematical series
of convergence for a function series, such as uniform convergence, pointwise convergence, and convergence almost everywhere. Each type of convergence corresponds
Function_series
Concept in General Topology
topology yields the topology of pointwise convergence; sequences of functions converge if and only if they converge at every point of S {\displaystyle
Box_topology
Generalization of a sequence of points
{\displaystyle \mathbb {R} ^{\mathbb {R} }} is identical to the topology of pointwise convergence. Let E {\displaystyle E} denote the set of all functions f : R →
Net_(mathematics)
Vector space of infinite sequences
norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space
Sequence_space
List of concrete topologies and topological spaces
uniform convergence. Compact-open topology Loop space Interlocking interval topology Modes of convergence (annotated index) Operator topologies Pointwise convergence
List_of_topologies
Sequence space that is Fréchet
topology of pointwise convergence. Thus the name coordinate space because a sequence in an FK-space converges if and only if it converges for each coordinate
FK-space
Infinite sum
of convergence of a series of functions is uniform convergence. A series converges uniformly in a set E {\displaystyle E} if it converges pointwise to
Series_(mathematics)
In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Ulisse Dini (1880). Dini's criterion states
Dini_criterion
Generalized function whose value is zero everywhere except at zero
associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere. If the initial η = η1 is itself smooth and compactly
Dirac_delta_function
Set of functions between two fixed sets
this context, this topology is also referred to as the topology of pointwise convergence. In algebraic topology, the study of homotopy theory is essentially
Function_space
Result in measure theory
theorem, in the form stated here, implies that almost everywhere pointwise convergence of the probability density functions of a sequence of μ {\displaystyle
Scheffé's_lemma
Kind of mathematical function
continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence. Real-valued functions encountered in applications
Measurable_function
Concept in mathematical analysis
series of a continuous function may fail to converge pointwise, in rather dramatic fashion. See convergence of Fourier series for further details. A precise
Dirichlet_kernel
Topological group structure arising in Fourier analysis
\mathbb {T} } . The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. This is the topology of the circle
Locally_compact_abelian_group
Branch of mathematics
measure theory, pointwise convergence of functions can be replaced with the notion of convergence almost everywhere, that is, convergence at every point
Mathematical_analysis
Function of two vectors linear in each argument
topology of bounded convergence; give all three the topology of compact convergence; give all three the topology of pointwise convergence. If E {\displaystyle
Bilinear_map
Mathematical tool to algorithmically solve equations
the point-wise convergence of { y n } n ∈ N {\displaystyle \{y_{n}\}_{n\in \mathbb {N} }} to y {\displaystyle y} implies the convergence of the associated
Numerical_method
Distribution function associated with the empirical measure of a sample
{\widehat {F}}_{n}(t)} is consistent. This expression asserts the pointwise convergence of the empirical distribution function to the true cumulative distribution
Empirical distribution function
Empirical_distribution_function
Theorems concerning stochastic processes
martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There
Doob's martingale convergence theorems
Doob's_martingale_convergence_theorems
Dual pair of vector spaces
{\displaystyle Y} is complete in the weak-* topology (i.e. the topology of pointwise convergence). Consequently, when the continuous dual space X ′ {\displaystyle
Dual_system
Type of mathematical space
x. The coarsest such topology, sometimes called the topology of pointwise convergence, is the product topology. With this topology, K is a compact topological
Compact_space
Mathematical representation in functional analysis
weak-* topology. This is the topology of pointwise convergence. A net {fk}k of elements of the spectrum of A converges to f if and only if for each x in A
Gelfand_representation
not the case. For the conclusion to hold, "pointwise convergence" must be replaced with "uniform convergence". It is not entirely clear that Cauchy's original
List_of_incomplete_proofs
Algebraic structure in linear algebra
of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are
Vector_space
Type of mathematical convergence in topology
mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated
Compact_convergence
Indicator function of rational numbers
(which are nonnegative, Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable. The Dirichlet
Dirichlet_function
Locally convex topology on function spaces
viewed as more natural, too, since it is simply the topology of pointwise convergence. The SOT topology also provides the framework for the measurable
Strong_operator_topology
Topological space that is homeomorphic to a metric space
R {\displaystyle \mathbb {R} } to itself, with the topology of pointwise convergence. The real line with the lower limit topology is not metrizable.
Metrizable_space
functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector
Topological_ring
Oscillatory error in Fourier series
exhibits pointwise convergence, but not uniform convergence. For a piecewise continuously differentiable (class C1) function, the Fourier series converges to
Gibbs_phenomenon
Notion in metric geometry
^{\infty }} norm, since ℓ ∞ {\displaystyle \ell ^{\infty }} convergence implies pointwise convergence. Thus T(X) is compact.) For any function g from X to R
Tight_span
Type of topological space
functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the
Normal_space
Averages of repeated trials converge to the expected value
that the sample mean of this sequence converges in probability to E[f(X,θ)]. This is the pointwise (in θ) convergence. A particular example of a uniform
Law_of_large_numbers
Curves whose limit does not preserve length
example showing that polyhedral surfaces that converge pointwise to a curved surface do not necessarily converge to its area, even when the vertices all lie
Staircase_paradox
Limit type in multivariable calculus
\left|a_{n,m}-b_{m}\right|<{\frac {\varepsilon }{3}}} . By the pointwise convergence, for any ε > 0 {\displaystyle \varepsilon >0} and n > N 1 {\displaystyle
Iterated_limit
Class of statistical estimators
The uniform convergence constraint is not necessarily required; an alternate set of assumptions is to instead consider pointwise convergence (in probability)
M-estimator
Mathematical measure space associated to a random walk
{K}}_{o}(\cdot ,y)} has a relatively compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A
Poisson_boundary
Theorem in measure theory
smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity. The strength of Lusin's
Lusin's_theorem
Particular kind of algebraic structure
operator norm) of a character is one. Equipped with the topology of pointwise convergence on A {\displaystyle A} (that is, the topology induced by the weak-*
Banach_algebra
Subset of all points that is bounded by some given point of a dual (in a dual pairing)
-valued functions on X {\displaystyle X} under the topology of pointwise convergence so when X # {\displaystyle X^{\#}} is endowed with the subspace
Polar_set
Mathematical theorem about the Fourier series
the proof. In fact, Fejér's theorem can be modified to hold for pointwise convergence. Modified Fejér's Theorem—Let f ∈ L 2 ( − π , π ) {\displaystyle
Fejér's_theorem
Chinese-American mathematician
Theorem, using novel techniques to solve Carleson's problem on pointwise convergence of solutions to the Schrödinger equation and solving the two-dimensional
Ruixiang_Zhang
In mathematics, vector space of linear forms
on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined
Dual_space
Relation among continuous functions
ƒn(x) = g(x − n). Then, ƒn converges pointwise to 0 but does not converge uniformly to 0. This criterion for uniform convergence is often useful in real
Equicontinuity
Function in mathematical analysis
under pointwise convergence. If f and g are bounded real-valued functions on the metric space X, with moduli respectively ω1 and ω2, then the pointwise product
Modulus_of_continuity
Function whose domain is the positive integers
functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average
Arithmetic_function
Every Boolean algebra is isomorphic to a certain field of sets
and so are clopen (both closed and open). This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. For
Stone's representation theorem for Boolean algebras
Stone's_representation_theorem_for_Boolean_algebras
Fair division problem
profiles which map to a specific allocation is a closed set under pointwise convergence. The following is proved for partners that assign positive utility
Utilitarian_cake-cutting
the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire space (set theory). Base A collection B of open sets
Glossary_of_general_topology
and ∞0. Pointwise convergence, Uniform convergence Absolute convergence, Conditional convergence Normal convergence Radius of convergence Integral test
List_of_real_analysis_topics
Branch of mathematics that studies dynamical systems
distribution of probabilities on the unit interval. More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the
Ergodic_theory
representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G. This
Algebraic_topology_(object)
On when a family of real, continuous functions has a uniformly convergent subsequence
the sequence {fn}, is pointwise bounded (or just bounded at a single point). Then there is a subsequence of the {fn} converging uniformly to a continuously
Arzelà–Ascoli_theorem
)} can be considered a topological group, given the topology of pointwise convergence, or equivalently, the subspace topology induced on Aut ( F ) {\displaystyle
Structural_Ramsey_theory
Haïm Brézis and Elliott Lieb. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983),
Brezis–Lieb_lemma
Type of topological vector space
bounded for the topology of pointwise convergence; H {\displaystyle H} is bounded for the topology of bounded convergence; H {\displaystyle H} is equicontinuous
Barrelled_space
converges in the weak-* topology σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} (also known as the topology of pointwise convergence)
Grothendieck_space
Mathematical concept
space of representations as a topological space with an appropriate pointwise convergence topology. More precisely, let n be a cardinal number and let Hn
Spectrum_of_a_C*-algebra
Average value of a random variable
} Furthermore, let X n → X {\displaystyle X_{n}\to X} pointwise. Then, the monotone convergence theorem states that lim n E [ X n ] = E [ X ] . {\displaystyle
Expected_value
Weak topology on function spaces
topology, or SOT, on B ( H ) {\displaystyle B(H)} is the topology of pointwise convergence. Because the inner product is a continuous function, the SOT is
Weak_operator_topology
Theorem
sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability
Skorokhod's representation theorem
Skorokhod's_representation_theorem
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POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
POINTWISE CONVERGENCE
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