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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 ,ρ 1 ), ( X 2 ,ρ 2 ) be metric spaces. On X 1 × X 2 , define d p (( x 1 ,x 2 ) , ( y 1 ,y 2 )) = ( ρ 1 ( x 1 ,y 1 ) p + ρ 2 ( x 2 ,y 2 ) p ) 1 /p for 1 ≤ p < ∞ max( ρ 1 ( x 1 ,y 1 ) ,ρ 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,ρ ) be a metric space, and put the d p metric on X × X . Prove that ρ : X × X → [0 , ∞ ) is uniformly continuous. (c) Let ( X,ρ ) 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 (ac) in Problem 8 on Homework 1....
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 Spring '10
 tao/analysis
 Math, Topology, Metric space, Fσ, Baire Category Theorem

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