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Unformatted text preview: p 6 = 2. 1 5. We have max ≤ i ≤ n  x i  p ≤ n X  x i  p , hence k x k ∞ ≤ k x k p . Also, n X  x i  p ≤ n max ≤ i ≤ n  x i  p , hence k x k ∞ ≥ n1 p k x k p . 6. Using the naming convention for constants in the previous problem, k x k q ≤ 1 c q k x k ∞ ≤ d p c q k x k p . Similarly, k x k q ≥ 1 d q k x k ∞ ≥ c p d q k x k p . 7. For z ∈ A d ( x , z ) ≤ d ( x , y ) + d ( y , z ) , hence inf z ∈ A d ( x , z ) ≤ d ( x , y ) + inf z ∈ A d ( y , z ) , which gives, by de±nition, d ( x , A ) ≤ d ( x , y ) + d ( y , A ) . Hence, d ( x , A )d ( y , A ) ≤ d ( x , y ) . Similarly d ( y , A )d ( x , A ) ≤ d ( x , y ) , hence  d ( x , A )d ( y , A )  ≤ d ( x , y ) . 2...
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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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