limit laws for uniform convergence

As might be expected, the usual (pointwise) limit laws — for instance, that the sum of limits is the limit of sums — have analogues for uniform limits. The laws for uniform limits are usually not mentioned in elementary textbooks, but they are useful in situations where having uniform convergenceMathworldPlanetmath is crucial, such as when working with infinite sums or products of holomorphic or meromorphic functions.

The uniform laws may be derived simply by reinterpreting the pointwise proofs from any calculus text, with some extra conditions. Some of these results are listed below.

In the following, X is a metric space, and Y is another metric space (with the appropriate operations defined), while fn:XY and gn:XY (with n) are functions that converge uniformly on X to f:XY and g:XY, respectively. (For example, X and Y may be , the complex plane.)

Theorem 1.

If Y is a normed vector spacePlanetmathPlanetmath, then fn+gn uniformly converge to f+g.

Theorem 2.

If Y is a Banach algebraMathworldPlanetmath, and both f(X) and g(X) are boundedPlanetmathPlanetmathPlanetmathPlanetmath, then fngn uniformly converge to fg.

Theorem 3.

Let Z be another metric space. Suppose that X is compactPlanetmathPlanetmath, Y is locally compact, and f:XY, and h:YZ are continuous functionsMathworldPlanetmath. Then hfn converge to hf uniformly.

Title limit laws for uniform convergence
Canonical name LimitLawsForUniformConvergence
Date of creation 2013-03-22 15:23:06
Last modified on 2013-03-22 15:23:06
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 4
Author stevecheng (10074)
Entry type Theorem
Classification msc 40A30