 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008-printed-variant/p/p3/e/solution

The residual is orthogonal to

q \in Π_{k}

, so Pythagoras gives

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

Taking nonnegative square roots proves

∥ q ∥ \leq ∥ f ∥

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