HW105 - -x ) + Z k < 1. Inductively, we may choose n 1...

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Analysis 1 Homework 05 1. Let Z be a closed subspace of the normed space X . Show that the quotient vector space X / Z carries a norm defined by the following rule: if x X then k x + Z k = inf {k x + z k : z Z } = d ( x , Z ) . Further, show that if X / Z and Z are complete then so is X itself. Remark : In the opposite direction, if X is complete then so are its closed subspace Z (of course) and its quotient X / Z . Solution 1. Let ( x n ) n be a Cauchy sequence in X . From k ( x q + Z ) - ( x p + Z ) k = k ( x q - x p ) + Z k 6 k x q - x p k it follows that the sequence ( x n + Z ) n in X / Z is Cauchy, hence convergent; say it converges to x + Z . Choose n 1 such that k ( x n - x ) + Z k < 1 whenever n > n 1 ; in particular, k ( x n 1
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Unformatted text preview: -x ) + Z k < 1. Inductively, we may choose n 1 < n 2 < ... such that if k = 1 , 2 , ... then k ( x n k-x ) + Z k < 1 / k . By definition of the quotient norm as an infimum, for each k there exists z k ∈ Z such that k x n k-x + z k k < 1 / k . Now ( x n k ) k is Cauchy and ( x n k-x + z k ) k is null, so ( z k ) k is Cauchy; say z k → z . Thus ( x n k ) k converges to x-z and so its parent (Cauchy) sequence ( x n ) n also converges to x-z . Done. 1...
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This note was uploaded on 07/08/2011 for the course MHF 3202 taught by Professor Larson during the Spring '09 term at University of Florida.

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