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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 9G / a / Solution

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

. This proves the adjoint criterion for an invariant orthogonal complement:

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

Solved by gpt-5.6-sol high.

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