hp - Metric Spaces Math 413 Honors Project 1 Metric Spaces...

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Unformatted text preview: Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X × X → R such that for all x, y, z ∈ X : i) d ( x, y ) = d ( y, x ); ii) d ( x, y ) = 0 if and only if x = y ; iii) d ( x, y ) ≤ d ( x, z ) + d ( z, y ) . If d is a metric on X we call ( X, d ) a metric space . We think of d ( x, y ) as the distance from x to y . Metric spaces arise in mathematics in many guises. Many of the basic properties of R that we will study in Math 413 are really properties of metric spaces and it is often useful to understand these ideas in full generality. We already know some natural examples of metric spaces. Exercise 1.2 [Euclidean metric] Suppose x = ( x 1 , . . . , x n ), y = ( y 1 , . . . , y n ) ∈ R n . Let d ( x , y ) = v u u t n X i =1 ( x i- y i ) 2 . a) For n = 1 show that d ( x, y ) = | y- x | is a metric on R . Note that d is also a metric on Q . In fact d is a metric on R n . The hard part is showing that iii) holds. For notational simplicity assume n = 2. b) Show that 2 xy ≤ x 2 + y 2 for any x, y ∈ R . c) (Schwartz Inequality) | x 1 y 1 + x 2 y 2 | ≤ p x 2 1 + x 2 2 p y 2 1 + y 2 2 [Hint: we may as well assume all the x i , y i ≥ 0.] 1 d) Use the Schwartz inequlality to show that p ( x 1 + y 1 ) 2 + ( x 2 + y 2 ) 2 ≤ p x 1 1 + x 2 2 + p y 2 1 + y 2 2 . e) Show that d is a metric on R 2 . [Hint: Use d) and the fact that x i- y i = ( x i- z i ) + ( z i- y i ).] There are other interesting examples. Exercise 1.3 [Discrete Spaces] Let X be any nonempty set. Define d ( x, y ) = n if x = y 1 otherwise . Prove that d is a metric. Exercise 1.4 [Taxi Cab Metric] For x , y ∈ R n define d ( x , y ) = X | x i- y i | . Prove that d is a metric. Why is this called the taxi cab metric in R 2 ? Exercise 1.5 [Sequence Spaces] Let A be any nonempty set and let Seq A be the set of all infinite sequences ( a 1 , a 2 , . . . ) where each a i ∈ A . If a = ( a 1 , a 2 , . . . ) and b = ( b 1 , b 2 , . . . ) are in Seq A define d ( a , b ) = ‰ if a n = b n for all n 1 n if n is least such that a n 6 = b n . Prove that d is a metric on Seq A ....
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This note was uploaded on 10/25/2011 for the course ECON 501 taught by Professor Zobuz during the Spring '11 term at Istanbul Technical University.

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hp - Metric Spaces Math 413 Honors Project 1 Metric Spaces...

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