hw1 - 5. Consider the Function f ( x ) = 1 1 + x 2 , x R ....

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Homework 1 – Math CS 120, Winter 2009 Due on Thursday, January 15th, 2009 1. Prove that the function k · k : R n -→ R deFned by k x k = max 1 i n | x i | is a norm. 2. Prove that the function k · k 1 : R n -→ R deFned by k x k 1 = n X i =1 | x i | is a norm. 3. The objective of this problem is to prove that the function k · k p : R n -→ R deFned by k x k p = p X i =1 | x i | p ! 1 /p , 1 < p < , is a norm. (a) Prove that x, y 0, and p 1, xy 1 p x p + 1 q y q , where 1 p + 1 q = 1 . Hint: Find max x 0 ± xy - 1 p x p ² . (b) Using the previous result, prove H¨ older’s inequality: n X i =1 | x i y i | ≤ k x k p k y k q , where 1 p + 1 q = 1 . This generalizes the Cauchy-Schwarz inequality, which corresponds to the case p = q = 2. 1
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2 (c) Prove that k · k p is a norm in R n . ( Hint: Split | x i + y i | p | x i || x i + y i | p - 1 + | y i || x i + y i | p - 1 , and apply H¨ older’s inequality to each term. (d) Prove that lim p →∞ k x k p = k x k x R n . 4. Prove, using the defnition oF continuity, that f ( x ) = x 1 / 3 , For x [0 , 1], is a continuous Function. Is it uniFormly continuous? Why?
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Unformatted text preview: 5. Consider the Function f ( x ) = 1 1 + x 2 , x R . Prove, using the defnition, that (a) lim x f ( x ) = 0 . (1) (b) f is uniFormly continuous. 6. Given the convergent sequences { a n } n N R and { b n } n N R , let a R and b R be the corresponding limits, i.e., lim n a n = a ; lim n b n = b. Prove, using the defnition oF limit, the Following statements: (a) lim n ( a n + b n ) = a + b . (b) lim n a n b n = ab . (c) IF b 6 = 0, lim n a n b n = a b . 7. Prove the Following statement: A Function f : I R is continuous at x I iF and only iF { x n } n N I such that x n x , it Follows that f ( x n ) f ( x )....
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hw1 - 5. Consider the Function f ( x ) = 1 1 + x 2 , x R ....

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