 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/p/p3/d/iii/solution

The decomposition

x = P_{n} x + (x - P_{n} x)

is orthogonal by (i). Therefore

∥ x ∥^{2} = ∥ P_{n} x ∥^{2} + ∥ x - P_{n} x ∥^{2}

. Substitute (ii) and rearrange to obtain

∥ x - P_{n} x ∥^{2} = ∥ x ∥^{2} - k = 1 \sum n ∣ ⟨ x, e_{k} ⟩ ∣^{2} .

In particular nonnegativity of the left side gives the finite Bessel inequality.

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