 Solution

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

Expand the norm of the orthonormal-basis expression. All cross terms vanish and

∥ e_{k} ∥ = 1

, so

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

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