02-math - Mathematical Appendix 1 Real Analysis...

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Unformatted text preview: Mathematical Appendix 1 Real Analysis Preliminaries First we need to build a mathematical apparatus. This section serves the purpose by providing a collection of definitions from Real Analysis and ex- amples demonstrating their application. 1 1.1 Metric spaces The key object in Real Analysis is a metric space. We will now build our understanding of what it is. Definition. Set is a collection of objects. Elements are individual objects belonging to a set. As an example, X ≡ { truth, lie } is a set. “Truth” and “lie” are X ’s elements. Now let’s specify several commonly used sets with standard no- tation. Natural numbers { , 1 , 2 , ... } comprise a set denoted by N . Real numbers comprise a set denoted by R . Set of n-dimensional vectors is de- noted by R n . Set of all continuous functions is denoted by C . Sometimes we will specify the set using the notation X ≡ { x : condition(x) is true } , e.g. X = { x : | x | lessorequalslant 1 } , X ≡ { 1 /n : n ∈ N } . Definition. Metric (distance function) is a function d : X × X → R + such that for any x, y, z ∈ X the following are true: a) d ( x, y ) = 0 if and only if x = y , b) ( Symmetry ) d ( x, y ) = d ( y, x ), c) ( Triangle Inequality ) d ( x, y ) lessorequalslant d ( x, z ) + d ( z, y ). Example. d ( x, y ) = | x- y | is a metric on the space of real numbers R as it trivially satisfies the properties (a)-(c) above. 1 If you want to dig deeper you should consult SLP. More on the mathematical side, by far the best Real Analysis text I encountered is Efe Ok’s “Real Analysis With Economic Applications”. Another good text on real analysis is “Infinite Dimensional Analysis: A Hitchhiker’s Guide” by Aliprantis and Border. 1 Example. d ( f, g ) = sup x ∈ X | f ( x )- g ( x ) | is a metric on the space of functions defined on the set X . 2 The first two properties are trivially true and the property (c) can be shown in the following way. Let f, g, h be functions defined on the set X . Then for any x ∈ X we have | f ( x )- g ( x ) | = | f ( x )- h ( x ) + h ( x )- g ( x ) | lessorequalslant | f ( x )- h ( x ) | + | h ( x )- g ( x ) | . In turn, sup x | f ( x )- g ( x ) | lessorequalslant sup x braceleftBig | f ( x )- h ( x ) | + | h ( x )- g ( x ) | bracerightBig lessorequalslant sup x | f ( x )- h ( x ) | + sup x | h ( x )- g ( x ) | = d ( f, h ) + d ( h, g ) , where we used the triangle inequality for the sup operator....
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This note was uploaded on 11/10/2011 for the course ECONOMICS 601 taught by Professor Viktortsyrennikov during the Spring '11 term at Cornell.

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02-math - Mathematical Appendix 1 Real Analysis...

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