hw1 - k k p : R n- R deFned by k x k p = p X i =1 | x i | p...

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Homework 1 – Math 118A, Fall 2009 Due on thursday, October 8, 2009 1. Let A 1 , A 2 , A 3 , . . . be subsets of a metic space. (a) If B n = n i =1 A i , prove that B n = n i =1 A i . (b) If B = i =1 A i , prove that B ⊃ ∪ i =1 A i . 2. Is every point of every open set E R 2 a limit point of E ? Answer the same question for closed sets in R 2 . 3. Let E denote the set of interior points of a set E : Prove that the complement of E is the closure of the complement of E . Do E and E always have the same interiors? Do E and E always have the same closures? 4. Consider an inFnite set X with the discrete metric: d ( x, y ) = ± 1 x 6 = y 0 x = y (1) Which subsets of the metric space ( X, d ) are open? Which are closed? Which are compact? 5. ±or x R and y R , deFne (a) d ( x, y ) = ( x - y ) 2 . (b) d ( x, y ) = sqrt | x - y | . (c) d ( x, y ) = | x 2 - y 2 | . (d) d ( x, y ) = | x - 2 y | . (e) d ( x, y ) = | x - y | 1+ | x - y | . Determine, for each of these, whether it is a metric or not. 1
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2 6. The objective of this problem is to prove that the function
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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 xy-1 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 Cauchy-Schwarz 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 | p-1 + | y i || x i + y i | p-1 , 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.

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hw1 - k k p : R n- R deFned by k x k p = p X i =1 | x i | p...

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