absolute convergence implies uniform convergence

Theorem 1.

Let $T$ be a topological space, $f$ be a continuous function from $T$ to $[0,\infty)$ , and let $\{f_{k}\}_{k=0}^{\infty}$ be a sequence of continuous functions from $T$ to $[0,\infty)$ such that, for all $x\in T$ , the sum $\sum_{k=0}^{\infty}f_{k}(x)$ converges to $f(x)$ . Then the convergence of this sum is uniform on compact subsets of $T$ .

Proof.

Let $X$ be a compact subset of $T$ and let $\epsilon$ be a positive real number. We will construct an open cover of $X$ . Because the series is assumed to converge pointwise, for every $x\in X$ , there exists an integer $n_{x}$ such that $\sum_{k=n_{x}}^{\infty}f_{k}(x)<\epsilon/3$ . By continuity, there exists an open neighborhood $N_{1}$ of $x$ such that $|f(x)-f(y)|<\epsilon/3$ when $y\in N_{1}$ and an open neighborhood $N_{2}$ of $x$ such that $\left|\sum_{k=0}^{n_{x}}f_{k}(x)-\sum_{k=0}^{n}f_{k}(y)\right|<\epsilon/3$ when $y\in N_{2}$ . Let $N_{x}$ be the intersection of $N_{1}$ and $N_{2}$ . Then, for every $y\in N$ , we have

$f(y)-\sum_{k=0}^{n_{x}}f_{k}(y)<|f(y)-f(x)|+\left|f(x)-\sum_{k=0}^{n_{x}}f_{k}% (x)\right|+\left|\sum_{k=0}^{n_{x}}f_{k}(x)-\sum_{k=0}^{n_{x}}f_{k}(y)\right|<\epsilon.$

In this way, we associate to every point $x$ an neighborhood $N_{x}$ and an integer $n_{x}$ . Since $X$ is compact, there will exist a finite number of points $x_{1},\ldots x_{m}$ such that $X\subseteq N_{x_{1}}\cup\cdots\cup N_{x_{m}}$ . Let $n$ be the greatest of $n_{x_{1}},\ldots,n_{x_{m}}$ . Then we have $f(y)-\sum_{k=0}^{n}f_{k}(y)<\epsilon$ for all $y\in X$ , so, the functions $f_{k}$ being positive, $f(y)-\sum_{k=0}^{h}f_{k}(y)<\epsilon$ for all $h\geq n$ , which means that the sum converges uniformly. ∎

Note: This result can also be deduced from Dini’s theorem, since the partial sums of positive functions are monotonically increasing.

Title	absolute convergence implies uniform convergence
Canonical name	AbsoluteConvergenceImpliesUniformConvergence
Date of creation	2013-03-22 18:07:27
Last modified on	2013-03-22 18:07:27
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 40A30