Uniform convergence

of a sequence of functions (mappings)

A property of a sequence $ f _ {n} : X \rightarrow Y $, where $ X $ is an arbitrary set, $ Y $ is a metric space, $ n = 1, 2 \dots $ converging to a function (mapping) $ f: X \rightarrow Y $, requiring that for every $ \epsilon > 0 $ there is a number $ n _ \epsilon $( independent of $ x $) such that for all $ n > n _ \epsilon $ and all $ x \in X $ the inequality

$$ \rho ( f ( x), f _ {n} ( x)) < \epsilon $$

holds. This is equivalent to

$$ \lim\limits _ {n \rightarrow \infty } \ \sup _ {x \in X } \ \rho ( f _ {n} ( x), f ( x)) = 0. $$

In order that a sequence $ \{ f _ {n} \} $ converges uniformly on a set $ X $ to a function $ f $ it is necessary and sufficient that there is a sequence of numbers $ \{ \alpha _ {n} \} $ such that $ \lim\limits _ {n \rightarrow \infty } \alpha _ {n} = 0 $, as well as a number $ n _ {0} $ such that for $ n > n _ {0} $ and all $ x \in X $ the inequality

$$ \rho ( f _ {n} ( x), f ( x)) \leq \alpha _ {n} $$

holds.

Example. The sequence $ \{ f _ {n} ( x) \} = \{ x ^ {n} \} $, $ n = 1, 2 \dots $ converges uniformly on any interval $ [ 0, a] $, $ 0 < a < 1 $, but does not converge uniformly on $ [ 0, 1] $.

A necessary and sufficient condition for uniform convergence that does not use the limit function is given by the Cauchy criterion for uniform convergence.

Properties of uniformly-convergent sequences.

1) If $ Y $ is a normed linear space and two sequences of mappings $ f _ {n} : X \rightarrow Y $ and $ g _ {n} : X \rightarrow Y $ converge uniformly on $ X $, then for any $ \lambda , \mu \in \mathbf C $ the sequence $ \{ \lambda f _ {n} + \mu g _ {n} \} $ also converges uniformly on $ X $.

2) If $ Y $ is a linear normed ring, if the sequence $ f _ {n} : X \rightarrow Y $, $ n = 1, 2 \dots $ is uniformly convergent on $ X $ and if $ g: X \rightarrow Y $ is a bounded mapping, then the sequence $ \{ gf _ {n} \} $ also converges uniformly on $ X $.

3) If $ X $ is a topological space, $ Y $ is a metric space and if a sequence of mappings $ f _ {n} : X \rightarrow Y $, continuous at $ x _ {0} \in X $, converges uniformly on $ X $ to $ f: X \rightarrow Y $, then $ f $ is also continuous at $ x _ {0} $, that is,

$$ \lim\limits _ {x \rightarrow x _ {0} } \ \lim\limits _ {n \rightarrow \infty } \ f _ {n} ( x) = \ \lim\limits _ {n \rightarrow \infty } \ f _ {n} ( x _ {0} ) = \ \lim\limits _ {n \rightarrow \infty } \ \lim\limits _ {x \rightarrow x _ {0} } \ f _ {n} ( x). $$

The condition of uniform convergence of the sequence $ \{ f _ {n} \} $ on $ X $ is essential in this result, in the sense that there are sequences of numerical functions, continuous on an interval, that converge at all points to a function that is not continuous on the interval in question. An example is $ f _ {n} ( x) = x ^ {n} $, $ n = 1, 2 \dots $ on $ [ 0, 1] $. Uniform convergence of a sequence of continuous functions is not a necessary condition for continuity of the limit function. However, if $ X $ is a compact set, $ Y $ is the set of real numbers $ \mathbf R $ and if all functions in a sequence of continuous functions $ f _ {n} : X \rightarrow \mathbf R $ simultaneously increase or decrease at all points $ x \in X $ and the sequence has a finite limit:

$$ \lim\limits _ {n \rightarrow \infty } f _ {n} ( x) = f ( x), $$

then in order that $ f $ be continuous on $ X $ it is necessary and sufficient that $ \{ f _ {n} \} $ converges uniformly on that set. Necessary, and simultaneously sufficient, conditions for the continuity of the limit of a sequence of continuous functions in general are given in terms of quasi-uniform convergence of the sequence.

4) If a sequence of Riemann- (Lebesgue-) integrable functions $ f _ {n} : [ a, b] \rightarrow \mathbf R $, $ n = 1, 2 \dots $ converges uniformly on $ [ a, b] $ to a function $ f: [ a, b] \rightarrow \mathbf R $, then this function is also Riemann- (respectively, Lebesgue-) integrable, for any $ x \in [ a, b] $ one has

$$ \tag{* } \lim\limits _ {n \rightarrow \infty } \ \int\limits _ { a } ^ { x } f _ {n} ( t) dt = \ \int\limits _ { a } ^ { x } f ( t) dt = \ \int\limits _ { a } ^ { x } \lim\limits _ {n \rightarrow \infty } f _ {n} ( t) dt, $$

and the convergence of the sequence $ \{ \int _ {a} ^ {x} f _ {n} ( t) dt \} $ to $ \int _ {a} ^ {x} f ( t) dt $ is uniform on $ [ a, b] $. Formula (*) has been generalized to the case of a Stieltjes integral. If, however, a sequence of integrable functions $ f _ {n} $, $ n = 1, 2 \dots $ on $ [ a, b] $ only converges at each point of the interval to an integrable function $ f $, then (*) need not hold.

5) If a sequence of continuously differentiable functions $ f _ {n} : [ a, b] \rightarrow \mathbf R $, $ n = 1, 2 \dots $ on $ [ a, b] $ converges at some point $ x _ {0} \in [ a, b] $ and if the sequence of derivatives $ \{ df _ {n} /dx \} $ converges uniformly on $ [ a, b] $, then the sequence $ \{ f _ {n} \} $ also converges uniformly on $ [ a, b] $, its limit is a continuously differentiable function on the interval and

$$ { \frac{d}{dx } } \lim\limits _ {n \rightarrow \infty } f _ {n} ( x) = \ \lim\limits _ {n \rightarrow \infty } \ \frac{df _ {n} ( x) }{dx } ,\ \ a \leq x \leq b. $$

Let $ X $ be a set and $ Y $ a metric space. A family of functions (mappings) $ f _ \alpha : X \rightarrow Y $, $ \alpha \in \mathfrak U $, with $ \mathfrak U $ a topological space, is said to be uniformly convergent as $ \alpha \rightarrow \alpha _ {0} \in \mathfrak U $ to the function (mapping) $ f: X \rightarrow Y $ if for every $ \epsilon > 0 $ there is a neighbourhood $ U ( \alpha _ {0} ) $ of $ \alpha _ {0} $ such that for all $ \alpha \in U( \alpha _ {0} ) $ and $ x \in X $ the inequality

$$ \rho ( f( x), f _ \alpha ( x)) < \epsilon $$

holds.

For uniformly-convergent families of functions there are properties similar to the above-mentioned properties of uniformly-convergent sequences of functions.

The concept of uniform convergence of mappings can be generalized to the case when $ Y $ is a uniform space, in particular, when $ Y $ is a topological group.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3]	J.L. Kelley, "General topology" , Springer (1975)

Comments

The theorem that a monotone sequence of continuous functions converges uniformly to its pointwise limit if this limit is continuous, is known as Dini's theorem.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)
[a2]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78