 Solution

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

(iii) implies (ii): the preliminary bounds show

sup_{k} ∥ P_{k} x ∥ < \infty

for every

x

P_{k}

gives

sup_{k} ∥ P_{k} ∥ < \infty

(iii) implies (iv): the normalization

∥ x_{n} ∥ = 1

gives

∣ φ_{n} (x) ∣ = ∥ (P_{n} - P_{n - 1}) x ∥ \leq (∥ P_{n} ∥ + ∥ P_{n - 1} ∥) ∥ x ∥.

Hence every coordinate functional is bounded.

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