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Unformatted text preview: 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 xy1 p x p . (b) Using the previous result, prove H olders 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 CauchySchwarz inequality, which corresponds to the case p = q = 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  p1 +  y i  x i + y i  p1 , and apply H olders inequality to each term. (d) Prove that lim p k x k p = k x k x R n ....
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This note was uploaded on 12/27/2011 for the course MATH 118a taught by Professor Stopple,j during the Fall '08 term at UCSB.
 Fall '08
 Stopple,J
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