AnalysisHW3 125A

AnalysisHW3 125A - MAT 125a, HW3 Solutions 1(a) Consider...

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Unformatted text preview: MAT 125a, HW3 Solutions 1(a) Consider the set R n of n-tuples of real numbers with d ( x , y ) = n X i =1 | x i- y i | . We wish to verify that d defines a metric on R n : (1) Since d ( x , y ) is a sum of non-negative numbers we know that d ( x , y ) 0 for all x , y R n . (2) If x = y , their components are equal, and d ( x , y ) = 0. Conversely, suppose that d ( x , y ) = 0. By the non-negativities of the absolute values in the sum, we must have that | x i- y i | = 0 for i = 1 ,...,n (if one of them were positive then the sum would be positive). This means x i = y i for i = 1 ,...,n , and x = y . (3) That d ( x , y ) = d ( y , x ) follows from the symmetry of the vanilla absolute value on R . (4) To prove the triangle inequality for this metric, recall the triangle inequality for the vanilla absolute value on R : | a- b | | a- c | + | c- b | for all real numbers a,b and c . For any points x , y , z R n we have d ( x , y ) = n X i =1 | x i- y i | n X i =1 ( | x i- z i | + | z i- y i | ) = d ( x , z ) + d ( z , y ) where the inequality is just the application of the vanilla triangle inequality n times....
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AnalysisHW3 125A - MAT 125a, HW3 Solutions 1(a) Consider...

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