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 AbsoluteConvergenceImpliesUniformConvergence 2013-03-22 18:07:27 2013-03-22 18:07:27 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Theorem msc 40A30