 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2008/p/3/e/solution

The best-approximation characterization gives

⟨ f - q, q ⟩ = 0

, since

q \in Π_{k}

. Consequently

∥ f ∥^{2} = ∥ (f - q) + q ∥^{2} = ∥ f - q ∥^{2} + ∥ q ∥^{2} \geq ∥ q ∥^{2} .

Taking nonnegative square roots proves

∥ q ∥ \leq ∥ f ∥

. Thus an orthogonal projection cannot increase the norm.

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