请输入您要查询的字词:

 

单词 LimitLawsForUniformConvergence
释义

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,thenfn+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:YZare continuous functionsMathworldPlanetmath. Then hfn converge to hfuniformly.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 11:22:19