Table of contents |
2 Topological reformulation 3 Theorems 4 Generalizations 5 History 6 Reference |
Suppose S is a set and fn : S -> R are real-valued functions for every natural number n. We say that the sequence (fn) converges uniformly with limit f : S -> R iff
Definition and comparison with pointwise convergence
Compare this to the concept of pointwise convergence: The sequence (fn) converges pointwise with limit f : S -> R iff
Given a topological space X, we can equip the space of real/complex functions over X with the uniform norm topology. Then, uniform convergence simply means convergence in the uniform norm topology.
If S is a real interval (or indeed any topological space), we can talk about the continuity of the functions fn and f. The following is the more important result about uniform continuity:
Topological reformulation
Theorems
If S is an interval and all the functions fn are differentiable and converge to a limit f, it is often desirable to differentiate the limit function f by taking the limit of the derivatives of fn. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable, and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance fn(x) = 1/n sin(nx) with uniform limit 0, but the derivatives do not approach 0. The precise statement covering this situation is as follows:
Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, one needs to require uniform convergence:
If S is a compact interval (or in general a compact topological space), and (fn) is a monotone increasing sequence (meaning fn(x) ≤ fn+1(x) for all n and x) of continuous functions with a pointwise limit f which is also continuous, then the convergence is necessarily uniform ("Dini's theorem").
One may straightforwardly extend the concept to functions S -> M, where (M, d) is a metric space, by replacing |fn(x) - f(x)| with d(fn(x), f(x)).
The most general setting is the uniform convergence of netss of functions S -> X, where X is a uniform space. We say that the net (fα) converges uniformly with limit f : S -> X iff
Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Fourier and Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence.
Riemann pointed to the need for distinguishing between absolutely
and conditionally convergent series by his Rearrangement Theorem. It shows that
it is possible to rearrange the terms of a conditionally cnvergent series so
that the derived series convergest to any desired limitGeneralizations
The above mentioned theorem, stating that the uniform limit of continuous functions is continuous, remains correct in these settings.History