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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 22I / b / i / Solution

Take

X = Y = ℓ^{2}

and define

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.

Solved by gpt-5.6-sol high.

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