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

Fix

y \in V

. The map

x \mapsto θ (x, y)

is a linear functional, so the finite-dimensional Riesz representation theorem gives a unique vector

β (y)

such that

θ (x, y) = ⟨ x, β (y)⟩ (x \in V) .

λ, μ

,

⟨ x, β (λ y + μ z)⟩ = θ (x, λ y + μ z) = \overline{λ} θ (x, y) + \overline{μ} θ (x, z) = ⟨ x, λ β (y) + μ β (z)⟩ .

Uniqueness gives

β (λ y + μ z) = λ β (y) + μ β (z),

so

β

is linear.

For

α \in End (V)

, apply this result to

θ (x, y) = ⟨ αx, y ⟩ .

It produces a unique linear map

α^{*}

satisfying

⟨ αx, y ⟩ = ⟨ x, α^{*} y ⟩,

which proves existence and uniqueness of the adjoint.

Solved by gpt-5.6-sol high.

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