HW101 - Analysis 1 Homework 01 1. Show that the space c 00...

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Unformatted text preview: Analysis 1 Homework 01 1. Show that the space c 00 of finitely-nonzero scalar sequences is dense in both 1 and c . Is it dense in ? Justify. 2. Show that if z = ( z n ) n N then d ( z,c ) = lim sup n | z n | . 3. Let z be a unit vector in c . Prove that there exist distinct unit vectors x and y in c such that z = 1 2 ( x + y ). Solutions 1. Let a 1 and > 0; say a = ( a n ) n N as usual and choose N so that n>N | a n | < . Define x = ( x n ) n N by x n = a n if n 6 N and x n = 0 if n > N ; then x c 00 and || x- a || 1 = X n N | x n- a n | = X n>N | a n | < . Let a c and > 0; say a = ( a n ) n N as usual and choose N so that sup n>N | a n | < . Define x = ( x n ) n N by x n = a n if n 6 N and x n = 0 if n > N ; then x c 00 and || x- a || = sup n N | x n- a n | = sup n>N | a n | < . The case is different. Let a = ( a n ) n N be defined by a n = 1 for all n . If x...
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HW101 - Analysis 1 Homework 01 1. Show that the space c 00...

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