 Solution

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

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

v \in \overline{X}

, choose

x \in X

with

∥ v - x ∥ < ε

. By part (i),

∥ A_{n} v ∥ \leq ∥ A_{n} (v - x) ∥ + ∥ A_{n} x ∥ \leq ε + ∥ A_{n} x ∥.

The last term tends to zero by part (ii). Taking the limit superior and then

ε ↓ 0

proves $A_{n} v \to 0$ for every $v \in \overline{X}$ .

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