 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-1/22h/c/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

⟨ x_{n}, x_{\infty} ⟩ ⟶ ⟨ x_{\infty}, x_{\infty} ⟩ = ∥ x_{\infty} ∥^{2} .

Using the inner product identity,

∥ x_{n} - x_{\infty} ∥^{2} = ∥ x_{n} ∥^{2} + ∥ x_{\infty} ∥^{2} - 2 Re ⟨ x_{n}, x_{\infty} ⟩ ⟶ ∥ x_{\infty} ∥^{2} + ∥ x_{\infty} ∥^{2} - 2∥ x_{\infty} ∥^{2} = 0.

Hence (ii) implies (i); this is the Radon-Riesz theorem in a Hilbert space.

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