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Unformatted text preview: Econ 4111 Professor: John Nachbar 11/20/08 Metric Spaces 1 Metric Spaces Basics. 1.1 Metric spaces. A metric space ( X,d ) consists of a set of points, X together with a distance function, or metric, d : X X R . The interpretation is that d ( a,b ) is the distance between a and b . To qualify as a distance function, d must satisfy three properties. 1. For any a,b X , d ( a,b ) 0 with d ( a,b ) = 0 iff a = b . 2. For any a,b X , d ( a,b ) = d ( b,a ). 3. For any a,b,c X , d ( a,c ) d ( a,b ) + d ( b,c ). (The triangle inequality .) The triangle inequality gets its name because, in standard geometry, if the three points a , b , and c form a triangle then the length of the side from a to c is less then the sum of the lengths of the other two sides. Think of X as fundamental while the metric d is a kind of overlay, like the grid on a map, that we add to help with our analysis. Any X has an infinity of possible metrics. At a minimum, metrics can differ because of units (inches rather than centimeters). But it is possible for different metrics to give different answers to questions like, Is a further from c than b is from c ? (I give an example of this below). Which metic we choose depends entirely on which is most helpful to us. For some spaces, notably X = R N , there is a default metric that is convenient for almost any application. For other spaces, such as variants of X = R , there is no default metric. 1.2 The space R N . The most familiar example of a metric space is R N , which is the space of points of the form ( x 1 ,...,x N ), where each x n R . For R N , the default metric is the Euclidean metric d E defined by for all a,b R N , d E ( a,b ) = q ( a- b ) ( a- b ) = s X n ( a n- b n ) 2 . In R , R 2 , or R 3 , the Euclidean metric corresponds to the everyday notion of physical distance. It is easiest to see this if I focus on the special case of measuring distance to the origin. Given x R N , define k x k = d E ( x, 0) . 1 Then k x k , called the Euclidean norm of x , measures the distance of x from the origin. For example, suppose that x = ( x 1 ,x 2 ) R 2 . Then, the distance from the origin to x is the length of the hypotenuse of the right triangle with short sides given by ( x 1 , 0) and (0 ,x 2 ). By the Pythagorean theorem, the length of this hypotenuse is q x 2 1 + x 2 2 . To be careful, I have to verify that d E is a metric. Theorem 1. The Euclidean metric on R N satisfies the three metric properties. Of the metric properties, only the triangle inequality requires any work. To prove that, I first establish an important inequality called the Cauchy-Schwartz inequality. Theorem 2 (Cauchy-Schwartz) . For any a,b R N , | a b | k a kk b k Proof of Theorem 2. If a b = 0 then the result is immediate. Suppose a b 6 = 0....
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This note was uploaded on 05/20/2010 for the course ECON 511 taught by Professor Johnnachbar during the Fall '08 term at Washington University in St. Louis.
- Fall '08