hmwk3 - MATH 202A HOMEWORK 3 This assignment is due...

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Unformatted text preview: MATH 202A HOMEWORK 3 This assignment is due Wednesday, Sept. 19, and deals primarily with compactness, the Baire Category theorem, and associated concepts. Problem 1 . Prove that a metric space X is totally bounded if and only if every sequence in X has a Cauchy subsequence. The following facts were used in lecture in the proof that if K 1 , K 2 are disjoint compact subsets of a metric space, then dist( K 1 ,K 2 ) > 0. This could be proved instead using sequences, avoiding the use of the following facts, but the following facts are nevertheless frequently used and worth knowing and understanding. Problem 2 . Let ( X 1 , 1 ), ( X 2 , 2 ) be metric spaces. On X 1 X 2 , define d p (( x 1 ,x 2 ) , ( y 1 ,y 2 )) = ( 1 ( x 1 ,y 1 ) p + 2 ( x 2 ,y 2 ) p ) 1 /p for 1 p < max( 1 ( x 1 ,y 1 ) , 2 ( x 2 ,y 2 )) for p = (a) Show that for all p , 1 p , d p defines a metric on X 1 X 2 and that they are all uniformly equivalent. You may use facts we have already proved regarding ` p d with d = 2. (b) Let ( X, ) be a metric space, and put the d p metric on X X . Prove that : X X [0 , ) is uniformly continuous. (c) Let ( X, ) be a metric space, and K 1 X , K 2 X be compact subsets. Prove that K 1 K 2 is compact. Picking up on the theme of the above problem, the following facts are also frequently applied. Problem 3 . Let X be a vector space with norm k k . (a) Prove that the map N : X [0 , ) is uniformly continuous, where N ( x ) = k x k . (b) Prove that the map A : X X X is uniformly continuous, where A ( x,y ) = x + y . (c) Prove that the map S : C X X is continuous, where C ( c,x ) = cx . (d) Prove that if K 1 , K 2 are compact subsets of X , then K 1 + K 2 = { x X : x 1 K 1 , x 2 K 2 such that x = x 1 + x 2 } is compact. (e) Prove that if U X is open and E X is an arbitrary subset, then U + E is open. (Hint: Write U + E as a union of open sets) We actually used the above problem (a-c) in Problem 8 on Homework 1....
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hmwk3 - MATH 202A HOMEWORK 3 This assignment is due...

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