Uniform convergence

of a sequence of functions (mappings)

A property of a sequence , where is an arbitrary set, is a metric space, converging to a function (mapping) , requiring that for every there is a number (independent of ) such that for all and all the inequality

holds. This is equivalent to

In order that a sequence converges uniformly on a set to a function it is necessary and sufficient that there is a sequence of numbers such that , as well as a number such that for and all the inequality

holds.

Example. The sequence , converges uniformly on any interval , , but does not converge uniformly on .

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 is a normed linear space and two sequences of mappings and converge uniformly on , then for any the sequence also converges uniformly on .

2) If is a linear normed ring, if the sequence , is uniformly convergent on and if is a bounded mapping, then the sequence also converges uniformly on .

3) If is a topological space, is a metric space and if a sequence of mappings , continuous at , converges uniformly on to , then is also continuous at , that is,

The condition of uniform convergence of the sequence on 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 , on . Uniform convergence of a sequence of continuous functions is not a necessary condition for continuity of the limit function. However, if is a compact set, is the set of real numbers and if all functions in a sequence of continuous functions simultaneously increase or decrease at all points and the sequence has a finite limit:

then in order that be continuous on it is necessary and sufficient that 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 , converges uniformly on to a function , then this function is also Riemann- (respectively, Lebesgue-) integrable, for any one has

(*)

and the convergence of the sequence to is uniform on . Formula (*) has been generalized to the case of a Stieltjes integral. If, however, a sequence of integrable functions , on only converges at each point of the interval to an integrable function , then (*) need not hold.

5) If a sequence of continuously differentiable functions , on converges at some point and if the sequence of derivatives converges uniformly on , then the sequence also converges uniformly on , its limit is a continuously differentiable function on the interval and

Let be a set and a metric space. A family of functions (mappings) , , with a topological space, is said to be uniformly convergent as to the function (mapping) if for every there is a neighbourhood of such that for all and the inequality

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 is a uniform space, in particular, when 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