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204Lecture32009web

# 204Lecture32009web - Economics 204 Lecture 3Wednesday...

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Economics 204 Lecture 3–Wednesday, July 29, 2009 Section 2.1, Metric Spaces and Normed Spaces Generalization of distance notion in R n Definition 1 A metric space is a pair ( X, d ), where X is a set and d : X × X R + , satisfying 1. x,y X d ( x, y ) 0 , d ( x, y ) = 0 x = y 2. x,y X d ( x, y ) = d ( y, x ) 3. ( triangle inequality ) x,y,z X d ( x, y ) + d ( y, z ) d ( x, z ) y x z Definition 2 Let V be a vector space over R . A norm on V is a function · : V R + satisfying 1. x V x 0 2. x V x = 0 x = 0 3. ( triangle inequality ) x,y V x + y x + y x x y 0 x + y y x y 1

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4. α R ,x V αx = | α | x A normed vector space is a vector space over R equipped with a norm. Theorem 3 Let ( V, · ) be a normed vector space. Let d : V × V R + be defined by d ( v, w ) = v w Then ( V, d ) is a metric space. Proof: We must verify that d satisfies all the properties of a metric. 1. d ( v, w ) = v w 0 d ( v, w ) = 0 v w = 0 v w = 0 ( v + ( w )) + w = w v + (( w ) + w ) = w v + 0 = w v = w 2. First, note that for any x V , 0 · x = (0 + 0) · x = 0 · x + 0 · x , so 0 · x = 0. Then 0 = 0 · x = (1 1) · x = 1 · x + ( 1) · x = x + ( 1) · x , so we have ( 1) · x = ( x ). d ( v, w ) = v w = | − 1 | v w = ( 1)( v + ( w )) = ( 1) v + ( 1)( w ) = v + w 2
= w + ( v ) = w v = d ( w, v ) 3. d ( u, w ) = u w = u + ( v + v ) w = u v + v w u v + v w = d ( u, v ) + d ( v, w ) Examples of Normed Vector Spaces E n : n -dimensional Euclidean space. V = R n , x 2 = | x | = n i =1 ( x i ) 2 V = R n , x 1 = n i =1 | x i | V = R n , x = max {| x 1 | , . . . , | x n |} C ([0 , 1]) , f = sup {| f ( t ) | : t [0 , 1] } C ([0 , 1]) , f 2 = 1 0 ( f ( t )) 2 dt 3

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C ([0 , 1]) , f 1 = 1 0 | f ( t ) | dt Theorem 4 (Cauchy-Schwarz Inequality) If v, w R n , then n i =1 v i w i 2 n i =1 v 2 i n i =1 w 2 i | v · w | 2 | v | 2 | w | 2 | v · w | | v || w | Read the proof in De La Fuente. The Cauchy-Schwarz Inequality is essential in proving the triangle inequality in E n . Note that v · w = | v || w | cos θ where θ is the angle between v and w : v w θ 0 Definition 5 Two norms · and · on the same vector space V are said to be Lipschitz-equivalent if m,M > 0 x V m x x M x Equivalently, m,M > 0 x V,x =0 m x x M Theorem 6 (Not in De La Fuente) All norms on R n are Lipschitz-equivalent.
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