# absolute convergence implies uniform convergence

###### Theorem 1.

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

###### Proof.

Let $X$ be a compact subset of $T$ and let $\u03f5$ 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 $$. By continuity, there exists an open neighborhood ${N}_{1}$ of $x$ such that $$ when $y\in {N}_{1}$ and an open neighborhood ${N}_{2}$ of $x$ such that $$ 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

$$ |

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 ${x}_{1},\mathrm{\dots}{x}_{m}$ such that $X\subseteq {N}_{{x}_{1}}\cup \mathrm{\cdots}\cup {N}_{{x}_{m}}$. Let $n$ be the greatest of ${n}_{{x}_{1}},\mathrm{\dots},{n}_{{x}_{m}}$. Then we have $$ for all $y\in X$, so, the functions ${f}_{k}$ being positive, $$ for all $h\ge n$, 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 |