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HW102 - Analysis 1 Homework 02 1(i Let z c0 be nonzero nd c...

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Analysis 1 Homework 02 1. (i) Let z c 0 be nonzero; find φ c * 0 with k φ k = 1 and φ ( z ) = k z k . (ii) Let z 1 be nonzero; find φ * 1 with k φ k = 1 and φ ( z ) = k z k . 2. For z = ( z n ) n =1 c 0 write φ ( z ) = X n =1 z n 2 n . Prove: (i) φ is a bounded linear functional of norm 1; (ii) if z c 0 is nonzero then | φ ( z ) | < k z k . Solutions 1. (i) As z c 0 there exists N such that | z N | = k z k > 0. The specific ‘coordinate’ linear functional φ N : c 0 : w 7→ w N clearly does the job in this case: if w c 0 then | φ N ( w ) | = | w N | 6 k w k while | φ N ( z ) | = | z N | = k z k . (ii) Not quite so straightforward. Seek: a such that k a k = 1 and T a ( z ) = k z k 1 - that is, n N a n z n = n N | z n | . Found: for each n let a n be any scalar of absolute value 1 such that a n z n = | z
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