Uniform limit theorem

Last updated October 04, 2024

In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous.

Statement

More precisely, let X be a topological space, let Y be a metric space, and let ƒ_n : X → Y be a sequence of functions converging uniformly to a function ƒ : X → Y. According to the uniform limit theorem, if each of the functions ƒ_n is continuous, then the limit ƒ must be continuous as well.

This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒ_n : [0, 1] → R be the sequence of functions ƒ_n(x) = xⁿ. Then each function ƒ_n is continuous, but the sequence converges pointwise to the discontinuous function ƒ that is zero on [0, 1) but has ƒ(1) = 1. Another example is shown in the adjacent image.

In terms of function spaces, the uniform limit theorem says that the space C(X, Y) of all continuous functions from a topological space X to a metric space Y is a closed subset of Y^X under the uniform metric. In the case where Y is complete, it follows that C(X, Y) is itself a complete metric space. In particular, if Y is a Banach space, then C(X, Y) is itself a Banach space under the uniform norm.

The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and ƒ_n : X → Y is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be uniformly continuous.

Proof

In order to prove the continuity of f, we have to show that for every ε > 0, there exists a neighbourhood U of any point x of X such that:

d_{Y}(f(x),f(y))<\varepsilon ,\qquad \forall y\in U

Consider an arbitrary ε > 0. Since the sequence of functions (f_n) converges uniformly to f by hypothesis, there exists a natural number N such that:

d_{Y}(f_{N}(t),f(t))<{\frac {\varepsilon }{3}},\qquad \forall t\in X

Moreover, since f_N is continuous on X by hypothesis, for every x there exists a neighbourhood U such that:

d_{Y}(f_{N}(x),f_{N}(y))<{\frac {\varepsilon }{3}},\qquad \forall y\in U

In the final step, we apply the triangle inequality in the following way:

{\begin{aligned}d_{Y}(f(x),f(y))&\leq d_{Y}(f(x),f_{N}(x))+d_{Y}(f_{N}(x),f_{N}(y))+d_{Y}(f_{N}(y),f(y))\\&<{\frac {\varepsilon }{3}}+{\frac {\varepsilon }{3}}+{\frac {\varepsilon }{3}}=\varepsilon ,\qquad \forall y\in U\end{aligned}}

Hence, we have shown that the first inequality in the proof holds, so by definition f is continuous everywhere on X.

Uniform limit theorem in complex analysis

There are also variants of the uniform limit theorem that are used in complex analysis, albeit with modified assumptions.

Theorem.^[1] Let $\Omega$ be an open and connected subset of the complex numbers. Suppose that $(f_{n})_{n=1}^{\infty }$ is a sequence of holomorphic functions $f_{n}:\Omega \to \mathbb {C}$ that converges uniformly to a function $f:\Omega \to \mathbb {C}$ on every compact subset of $\Omega$ . Then $f$ is holomorphic in $\Omega$ , and moreover, the sequence of derivatives $(f'_{n})_{n=1}^{\infty }$ converges uniformly to $f'$ on every compact subset of $\Omega$ .

Theorem.^[2] Let $\Omega$ be an open and connected subset of the complex numbers. Suppose that $(f_{n})_{n=1}^{\infty }$ is a sequence of univalent^[3] functions $f_{n}:\Omega \to \mathbb {C}$ that converges uniformly to a function $f:\Omega \to \mathbb {C}$ . Then $f$ is holomorphic, and moreover, $f$ is either univalent or constant in $\Omega$ .

Notes

↑ Theorems 5.2 and 5.3, pp.53-54 in E. M. Stein and R.Shakarchi's Complex Analysis.
↑ Section 6.44, pp.200-201 in E. C. Titchmarsh's The Theory of Functions. Titchmarsh uses the terms 'simple' and 'schlicht' (function) in place of 'univalent'.
↑ Univalent means holomorphic and injective.

Related Research Articles

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

In complex analysis, the Riemann mapping theorem states that if $is a non-empty simply connected open subset of the complex number plane which is not all of, then there exists a biholomorphic mapping from onto the open unit disk$

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

<span class="mw-page-title-main">Uniform continuity</span> Uniform restraint of the change in functions

In mathematics, a real function $of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number, then there is a positive real number such that at any and in any function interval of the size .$

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions $converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number, a number can be found such that each of the functions differs from by no more than at every point in . Described in an informal way, if converges to uniformly, then how quickly the functions approach is "uniform" throughout in the following sense: in order to guarantee that differs from by less than a chosen distance, we only need to make sure that is larger than or equal to a certain, which we can find without knowing the value of in advance. In other words, there exists a number that could depend on but is independent of, such that choosing will ensure that for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find such that, for that particular, falls within of whenever .$

In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function $where U is an open subset of ⁠ ⁠ that satisfies Laplace's equation, that is, everywhere on U . This is usually written as or$

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 cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function.

In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

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 interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

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, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911.

In mathematics, a real or complex-valued function $f$ on $d$ -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants $C \geq 0$ , $α > 0$ , such that $for all x and y in the domain of f . More generally, the condition can be formulated for functions between any two metric spaces. The number is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.$

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in L^p in terms of convergence in measure and a condition related to uniform integrability.

In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rⁿ and Tⁿ through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L² is evident because the Fourier transform converts them into multiplication operators. Continuity on L^p spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on L^p spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.

In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hölder spaces, L^p spaces and Sobolev spaces. In the case of L² spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint.

In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS $with out any need to extend definitions from real/complex-valued functions to -valued functions.$

References

James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

E. M. Stein, R. Shakarchi (2003). Complex Analysis (Princeton Lectures in Analysis, No. 2), Princeton University Press.

E. C. Titchmarsh (1939). The Theory of Functions, 2002 Reprint, Oxford Science Publications.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Theorems 5.2 and 5.3, pp.53-54 in E. M. Stein and R.Shakarchi's Complex Analysis.

[2] Section 6.44, pp.200-201 in E. C. Titchmarsh's The Theory of Functions. Titchmarsh uses the terms 'simple' and 'schlicht' (function) in place of 'univalent'.

[3] Univalent means holomorphic and injective.

[1]

[2]

[3]