lecture6 - Lecture 6 Metric Spaces ECON 7250 VI - 2 Metric...

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Unformatted text preview: Lecture 6 Metric Spaces ECON 7250 VI - 2 Metric Spaces • A metric space is a space X endowed with a metric d . • A function d : X × X → R + is a metric iff it satisfies the following properties: – For all x , y ∈ X , d ( x , y ) = 0 iff x = y. (positive definiteness) – For all x , y ∈ X , d ( x , y ) = d ( y , x ). (symmetry) – For all x , y , z ∈ X , d ( x , z ) ≤ d ( x , y ) + d ( y , z ). (Triangle Inequality) ECON 7250 VI - 3 Examples of Metrics • R n endowed with the Euclidean metric d 2 ( x , y ) = | x- y | is the most common metric space. • There are infinitely many other metrics on R n . • Two other common metrics on R n : – d ∞ ( x , y ) = max i {| x i- y i |} – ∑ =- = n i i i y x y x d 1 1 ) , ( ECON 7250 VI - 4 Open Balls and Open Sets • If ( X , d ) is a metric space, we define for x ∈ X and r > 0, the open ball B ( x , r ) ≡ { y ∈ X : d ( x , y ) < r }. • U ⊆ X is called an open set iff, for every x ∈ U , there exists r > 0 such that B ( x , r ) ⊆ U . • An open ball is an open set. ECON 7250 VI - 5 Properties of Open Sets • Let ( X , d ) be a metric space. 1. Both X and ∅ are open. 2. Arbitrary unions of open sets are open. 3. Finite intersections of open sets are open. • Arbitrary intersections of open sets need not be open. ECON 7250 VI - 6 Interior Points • Let A ⊆ X and x ∈ A ....
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This note was uploaded on 01/09/2011 for the course ECON 7140 taught by Professor Kutler during the Spring '10 term at Utah Valley University.

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lecture6 - Lecture 6 Metric Spaces ECON 7250 VI - 2 Metric...

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