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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 22H / c / ii / Solution

Assume (ii). Weak convergence gives

⟨ 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.

Solved by gpt-5.6-sol high.

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