 Solution

ID: past-exam-of-the-functional-analysis-and-banach-and-c-algebras-course-of-the-university-of-oxford/2009/a/3/ii/solution

(ii) implies (i): together with the preliminary estimate, (ii) gives equivalent norms,

∥ x ∥ \leq ∥ x ∥^{'} \leq C ∥ x ∥.

An equivalent norm on a Banach space is complete: a primed-norm Cauchy sequence has an original-norm limit, and the upper bound makes convergence hold also in the primed norm. (ii) implies (iii) immediately because

∥ P_{k} ∥ \leq C

for every

k

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