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lecture15

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

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Lecture 15 We restate the partition of unity theorem from last time. Let { U α : α I } be a collection of open subsets of R n such that ± U = U α . (3.169) α I Theorem 3.30. There exist functions f i ⊆ C 0 ( U ) such that 1. f 1 0 , 2. supp f i U α , for some α , 3. For every p U , there exists a neighborhood U p of p such that U p supp f i = φ 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 α : α I } .” Let us look at some typical applications of partitions of unity. The first application is to improper integrals. Let φ : U R be a continuous map, and suppose φ (3.170) U is well-defined. Take a partition of unity f i = 1. The function f i φ is continuous and compactly supported, so it bounded. Let supp f i Q i for some rectangle Q i .

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

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