This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 CauchySchwartz inequality. Theorem 2 (CauchySchwartz) . 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....
View
Full
Document
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
 JohnNachbar

Click to edit the document details