# 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)