 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-3/24j/b/solution

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

L^{p}

norm with

1 \leq p \leq \infty

. Apply it to

f_{n} = (f_{n} - f) + f

and then to

f = (f - f_{n}) + f_{n}

. The two bounds combine to

∥ f_{n} ∥_{p} - ∥ f ∥_{p} \leq ∥ f_{n} - f ∥_{p} ⟶ 0.

Thus norm convergence implies convergence of the norms themselves.

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