204Lecture32009

204Lecture32009 - Economics 204 Lecture 3Wednesday, July...

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Unformatted text preview: Economics 204 Lecture 3Wednesday, July 29, 2009 Revised 7/29/09, Revisions Indicated by ** and Sticky Notes 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 ) , 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 k k : V R + satisfying 1. x V k x k 2. x V k x k = 0 x = 0 3. ( triangle inequality ) x,y V k x + y k k x k + k y k x x % & y x + y y & % x y 1 4. R ,x V k x k = | |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 defined by d ( v, w ) = k v w k Then ( V, d ) is a metric space. Proof: We must verify that d satisfies all the properties of a metric. 1. d ( v, w ) = k v w k d ( v, w ) = 0 k v w k = 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 ) = k v w k = | 1 |k v w k = k ( 1)( v + ( w )) k = k ( 1) v + ( 1)( w ) k 2 = k v + w k = 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 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 ( f ( t )) 2 dt C ([0 , 1]) , k f k 1 = Z 1 | f ( t ) | dt 3 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...
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204Lecture32009 - Economics 204 Lecture 3Wednesday, July...

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