ECEN601-Chapter2

ECEN601-Chapter2 - Chapter 2 Metric Spaces and Topology 2.1...

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Chapter 2 Metric Spaces and Topology 2.1 Metric Spaces A metric space is a set that has a well-defned “distance” between any two ele- ments oF the set. Mathematically, the notion oF a metric spaceabstractsaFewbasic properties oF Euclidean space. ±ormally, a metric space ( X,d ) is a set X and a Function d that is a metric on X . Defnition 2.1.1. A metric on a set X is a function d : X × X R that satisFes the following properties, 1. d ( x,y ) 0 x,y X ;equalityholdsifandonlyif x = y 2. d ( x,y )= d ( y,x ) x,y X 3. d ( x,y )+ d ( y,z ) d ( x,z ) x,y,z X . Example 2.1.2. The set of real numbers equipped with the metric of absolute dis- tance d ( x,y )= | x - y | deFnes the standard metric space of real numbers R . Example 2.1.3. Given x =( x 1 ,...,x n ) ,y =( y 1 ,...,y n ) R n ,the Euclidean metric d on R n is deFned by the equation d ( x ,y ) = ± ( x 1 - y 1 ) 2 + ··· +( x n - y n ) 2 . As implied by its name, the function d deFned above is a metric. 19
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20 CHAPTER 2. METRIC SPACES AND TOPOLOGY P2 .1 .4 . Let x =( x 1 ,...,x n ) ,y =( y 1 ,...,y n ) R n and consider the function ρ given by ρ ( x ,y ) =max {| x 1 - y 1 | ,..., | x n - y n |} . Show that ρ is a metric. P2.1 .5 . Let X be a metric space with metric d .DeFne ¯ d : X × X R by ¯ d ( x,y )=m in { d ( x,y ) , 1 } . Show that ¯ d is also a metric. Let ( X,d ) be a metric space. Then, elements of X are called points and the number d ( x,y ) is called the distance between x and y .Le t $> 0 and consider the set B d ( x,$ )= { y X | d ( x,y ) <$ } .Th i sse ti sca l ledthe d -open ball of radius $ centered at x . P2 .1 .6 . Suppose a B d ( x,$ ) with $> 0 .S h owt h a tt h e r ee x i s t sa d -open ball centered at a of radius δ ,say B d ( a,δ ) ,thatiscontainedin B d ( x,$ ) . One of the main beneFts of having a metric is that it provides some notion of “closeness” between points in a set. This alows one to discusss limits, convergence, open sets, and closed sets. Defnition 2.1.7. A sequence of elements from a set X is an inFnite list x 1 ,x 2 ,... where x i X for all i N o rm a l l y ,as e q u e n c e i se q u i v a l e n t t oa f u n c t i o n f : N X where x i = f ( i ) for all i N . Defnition 2.1.8. Consider a sequence x 1
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2.1. METRIC SPACES 21 Proof. Since x 1 ,x 2 ,...d -converges to some x ,thereisan N ,forany $> 0 ,such that d ( x,x n ) < $/ 2 for all n>N .Thetr iang leinequa l i tyfor d ( x m ,x n ) shows that, for all m,n > N , d ( x m ,x n ) d ( x m ,x )+ d ( x,x n )
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ECEN601-Chapter2 - Chapter 2 Metric Spaces and Topology 2.1...

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