 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4/22f/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

∥ U f ∥^{2} = ⟨ U^{*} U f, f ⟩ = ∥ f ∥^{2} .

Thus it is an isometry, and every power has norm one. The triangle inequality gives

∥ A_{n} f ∥ \leq \frac{1}{n} \sum_{k = 0}^{n - 1} ∥ U^{k} f ∥ = ∥ f ∥, ∥ A_{n} ∥ \leq 1.

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