 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-3/9g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Suppose first that

α (U) \subseteq U

. If

y \in U^{⊥}

and

u \in U

, then

⟨ u, α^{*} y ⟩ = ⟨ αu, y ⟩ = 0.

Thus

α^{*} (U^{⊥}) \subseteq U^{⊥}

Conversely, assume that latter inclusion. For

u \in U

and

y \in U^{⊥}

⟨ αu, y ⟩ = ⟨ u, α^{*} y ⟩ = 0.

Hence

αu \in (U^{⊥})^{⊥} = U

α (U) \subseteq U ⟺ α^{*} (U^{⊥}) \subseteq U^{⊥} .

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