assignment2

assignment2 - , ) and nd lim p 0+ f ( p ). Hint: Compute f...

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MATH 255 ASSIGNMENT 2 This assignment is due in class on Monday, January 24 Problems Please justify carefully your answers. 1. [10 points] Let X be the collection of all sequences of positive integers. If x = ( n j ) j =1 and y = ( m j ) j =1 are two elements of X , set k ( x,y ) = inf { j : n j 6 = m j } and d ( x,y ) = ( 0 if x = y 1 k ( x,y ) if x 6 = y. Prove that d is a metric on X . 2. [10 points] For x,y R , set d ( x,y ) = arctan | x - y | . Show that d is a metric on R . 3. [15 points] Let M ( n, R ) be the vector space of all n × n real matrices. Let 1 p < be given. For A = [ a ij ] M ( n, R ), set k A k = n X i =1 n X j =1 | a ij | p ! 1 /p . (1) Prove that k · k is a norm on M ( n, R ). (2) Suppose that p = 2. Show that k AB k ≤ k A kk B k . 4. [20 points] Let x = ( x 1 , ··· ,x n ) R n . For 0 < p < set k x k p = n X k =1 | x k | p ! 1 /p . 1
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Recall that k x k = max 1 k n | x k | . (1) Prove that lim p →∞ k x k p = k x k . (2) Suppose that n 2 and that x is such that x k 6 = 0 for all k . Prove that the function f ( p ) = k x k p is strictly decreasing on (0
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Unformatted text preview: , ) and nd lim p 0+ f ( p ). Hint: Compute f ( p ) using that f ( p ) = e 1 p ln ( P n k =1 | x k | p ) (1) 5. [20 points] Let x = ( x 1 , ,x n ) R n and g ( p ) = 1 n n X k =1 | x | p k ! 1 /p . (1) Using H olders inequality show that g ( p ) is increasing on (0 , ). (2) Find lim p g ( p ). (3) Suppose that x k 6 = 0 for all k . Show that lim p 0+ g ( p ) = ( | x 1 || x 2 || x n | ) 1 /n , (4) Show that (1) and (3) imply the inequality of geometric and arithmetic means, that is, ( | x 1 || x 2 || x n | ) 1 /n | x 1 | + | x 2 | + + | x n | n . 2...
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assignment2 - , ) and nd lim p 0+ f ( p ). Hint: Compute f...

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