lecture15

lecture15 - Lecture 15 We restate the partition of unity...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 15 We restate the partition of unity theorem from last time. Let { U : I } be a collection of open subsets of R n such that U = U . (3.169) I Theorem 3.30. There exist functions f i C ( U ) such that 1. f 1 , 2. supp f i U , for some , 3. For every p U , there exists a neighborhood U p of p such that U p supp f i = for all i > N p , 4. f i = 1 . Remark. Property (4) makes sense because of property (3), because at each point it is a finite sum. Remark. A set of functions satisfying property (4) is called a partition of unity . Remark. Property (2) can be restated as the partition of unity is subordinate to the cover { U : I } . Let us look at some typical applications of partitions of unity. The first application is to improper integrals. Let : U R be a continuous map, and suppose (3.170) U is well-defined. Take a partition of unity f i = 1. The function f i is continuous and compactly supported, so it bounded. Let supp f i Q i for some rectangle Q i ....
View Full Document

Page1 / 3

lecture15 - Lecture 15 We restate the partition of unity...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online