 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/10/a/i/solution

For

n \geq 1

, use

sin^{2} (n x) = [1 - cos (2 n x)] /2

⟨ f_{n}, f_{n} ⟩ = \int_{- π}^{π} sin^{2} (n x) d x = π (n \geq 1) .

The stated nonnegative index range also includes

n = 0

. Since

f_{0}

⟨ f_{0}, f_{0} ⟩ = 0

, so it must be removed from any proposed basis.

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