 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-i/4/b/ii/solution

f, g

U

⟨ f, h ⟩ ⟨ g, h ⟩ = ⟨ f, x^{2} ⟩ - ⟨ f, x^{2} ⟩ ⟨ f, f ⟩ - ⟨ g, x^{2} ⟩ ⟨ f, g ⟩ = 0, = ⟨ g, x^{2} ⟩ - ⟨ f, x^{2} ⟩ ⟨ g, f ⟩ - ⟨ g, x^{2} ⟩ ⟨ g, g ⟩ = 0.

Together with

⟨ f, g ⟩ = 0

, these identities prove pairwise orthogonality. The construction subtracts the orthogonal projection onto a finite-dimensional subspace

U

from

x^{2}

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