204Lecture32009web - Economics 204 Lecture 3Wednesday, July...

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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 DeFnition 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 DeFnition 2 Let V be a vector space over R .A norm on V is a function k·k : V R + satisfying 1. x V k x k≥ 0 2. x V k x k =0 x 3. ( triangle inequality ) x,y V k x + y k≤k x k + k y k x x y 0 x + y y &% x y 1
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4. α R ,x V k αx k = | α |k x k A normed vector space is a vector space over R equipped with a norm. Theorem 3 Let ( V, k·k ) be a normed vector space. Let d : V × V R + be defned by d ( v,w )= k v w k Then ( V,d ) is a metric space. Proof: We must verify that d satisFes all the properties of a metric. 1. d ( k v w k≥ 0 d ( )=0 ⇔k v w k =0 v w ( v +( w )) + w = w v +(( w )+ w w v +0= w v = w 2. ±irst, note that for any x V ,0 · x =(0+0 ) · x · x +0 · x ,so0 · x . Then0=0 · x = (1 1) · x =1 · x 1) · x = x 1) · x ,sowehave( 1) · x =( x ). d ( k v w k = |− 1 |k v w k = k ( 1)( v w )) k = k ( 1) v 1)( w ) k = k− v + w k 2
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= k w +( v ) k = k w v k = d ( w, v ) 3. d ( u, w )= k u w k = k u v + v ) w k = k u v + v w k ≤k u v k + k v w k = d ( u, v )+ d ( v,w ) Examples of Normed Vector Spaces E n : n -dimensional Euclidean space. V = R n , k x k 2 = | x | = v u u t n X i =1 ( x i ) 2 V = R n , k x k 1 = n X i =1 | x i | V = R n , k x k =max {| x 1 | ,..., | x n |} C ([0 , 1]) , k f k =sup {| f ( t ) | : t [0 , 1] } C ([0 , 1]) , k f k 2 = s Z 1 0 ( f ( t )) 2 dt 3
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C ([0 , 1]) , k f k 1 = Z 1 0 | f ( t ) | dt Theorem 4 (Cauchy-Schwarz Inequality) If v,w R n ,then n X i =1 v i w i ! 2 n X i =1 v 2 i ! n X 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 : vw - θ % 0 Defnition 5 Two norms k·k and 0 on the same vector space V are said to be Lipschitz-equivalent if m,M > 0 x V m k x k≤k x k 0 M k x k Equivalently, m,M > 0 x V,x 6 =0 m k x k 0 k
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at University of California, Berkeley.

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204Lecture32009web - Economics 204 Lecture 3Wednesday, July...

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