Question 2. [2, 5, 4; 3 marks] (a) Consider the normed spaces (p,|| . ||p) where 1 < p <00. Prove that the canonical basis B := {en\n\ basis as follows: 1,...,} is a Schauder (i) Prove that the set 3 is linearly independent. (ii) Prove that ∞ X = - Σxiei i=1 = (iii) Prove that for loo, oo) that there is no Schauder basis. (HINT: Use a counterexample.) (b) Consider the normed spaces (co, ) of all sequences with limit equal 0. Prove that the canonical basis 3:= {enn = 1,...,} is a Schauder basis for co.

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