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# MetricSpace - Econ 4111 Professor John Nachbar Metric...

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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

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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. For any λ 0, 0 ≤ k a - λb k 2 = k a k 2 - 2 λ ( a · b ) + λ 2 k b k 2 Since a · b 6 = 0, I can define λ = k a k 2 a · b .
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MetricSpace - Econ 4111 Professor John Nachbar Metric...

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