homework5 2010

homework5 2010 - R 2 under these metrics b If S ⊆ R n is...

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ECN/APEC 7240 Spring 2010 Homework 5 Due 3/1/10 1) Let x , y R n . We define the d 1 , d 2 , and d metrics such that . max ) , ( ) ( ) , ( ) , ( 2 / 1 1 2 2 1 1 i i i n i i i n i i i y x y x d y x y x d y x y x d - = - = - = = = Note that the d 2 metric corresponds to our usual notion of Euclidean distance. a) What is the shape of an open ball in
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Unformatted text preview: R 2 under these metrics? b) If S ⊆ R n is open with respect to d 2 , is it open with respect to d 1 or d ∞ ? 2) Argue that || f || ∞ = sup{| f ( x )| : x ∈ Ω } defines a norm on B ( Ω ), where Ω is any set....
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