absolute convergence implies uniform convergence

Theorem 1.

Let T be a topological spaceMathworldPlanetmath, f be a continuous functionMathworldPlanetmathPlanetmath from T to [0,), and let {fk}k=0 be a sequence of continuous functions from T to [0,) such that, for all xT, the sum k=0fk(x) convergesPlanetmathPlanetmath to f(x). Then the convergence of this sum is uniform on compact subsets of T.


Let X be a compact subset of T and let ϵ be a positive real number. We will construct an open cover of X. Because the series is assumed to converge pointwisePlanetmathPlanetmath, for every xX, there exists an integer nx such that k=nxfk(x)<ϵ/3. By continuity, there exists an open neighborhood N1 of x such that |f(x)-f(y)|<ϵ/3 when yN1 and an open neighborhood N2 of x such that |k=0nxfk(x)-k=0nfk(y)|<ϵ/3 when yN2. Let Nx be the intersectionMathworldPlanetmath of N1 and N2. Then, for every yN, we have


In this way, we associate to every point x an neighborhood Nx and an integer nx. Since X is compact, there will exist a finite number of points x1,xm such that XNx1Nxm. Let n be the greatest of nx1,,nxm. Then we have f(y)-k=0nfk(y)<ϵ for all yX, so, the functions fk being positive, f(y)-k=0hfk(y)<ϵ for all hn, 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