 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/22i/b/i/solution

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

Take

X = Y = ℓ^{2}

S (x_{1}, x_{2}, \dots) = (x_{1}, \frac{x _{2}}{2}, \frac{x _{3}}{3}, \dots) .

This is bounded and injective. Its inverse on

im S

is not continuous: for the standard unit vectors

e_{n}

∥ S e_{n} ∥_{2} = \frac{1}{n} ⟶ 0, ∥ S^{- 1} (S e_{n}) ∥_{2} = ∥ e_{n} ∥_{2} = 1.

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