absolute convergence implies uniform convergence
Theorem 1.
Let be a topological space, be a continuous function
from to , and
let be a sequence of continuous functions from to
such that, for all , the sum converges
to . Then
the convergence of this sum is uniform on compact subsets of .
Proof.
Let be a compact subset of and let be a positive real number. We will
construct an open cover of . Because the series is assumed to converge pointwise, for
every , there exists an integer such that . By continuity, there exists an open neighborhood of such that when and an open neighborhood of such that when .
Let be the intersection
of and . Then, for every , we have
In this way, we associate to every point an neighborhood and an integer . Since is compact, there will exist a finite number of points such that . Let be the greatest of . Then we have for all , so, the functions being positive, for all , 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 |
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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 |