HW107 - Analysis 1 Homework 07 1. True or False: The...

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Analysis 1 Homework 07 1. True or False: The sup-norm k • k is complete on 1 . 2. True or False: C [0 , 1] is isometrically isomorphic to a closed subspace of . 3. True or False: If Z is a closed subspace of the Banach space X then for each a X there exists a unique z Z such that k a - z k = d ( a , Z ). 4. True or False: If x 1 and x 2 are vectors of equal norm in the Banach space X then there exists an isometric isomorphism T : X X such that T ( x 1 ) = x 2 . Solutions 1. F. An indirect proof: note first that if x 1 then k x k 6 k x k 1 and that k • k 1 is complete, whence it would follow (as a consequence of the Banach Isomorphism Theorem) that if k • k were complete then there would exist K such that k • k 1 6 K k • k ; however, Archimedes prohibits this, for k e 1 + ··· + e n k = 1 and yet k e 1 + ··· + e n k 1 = n . A direct proof: the sequence ( n i = 1 e i / i ) n = 1 in ( c 00 ) 1 is plainly k • k
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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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