 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/p/p3/b/solution

Expand using the inner product. The two cross terms cancel:

∥ x + y ∥^{2} = ∥ x ∥^{2} + ∥ y ∥^{2} + 2ℜ ⟨ x, y ⟩, ∥ x - y ∥^{2} = ∥ x ∥^{2} + ∥ y ∥^{2} - 2ℜ ⟨ x, y ⟩ .

Adding proves

∥ x - y ∥^{2} + ∥ x + y ∥^{2} = 2 (∥ x ∥^{2} + ∥ y ∥^{2})

, for real or complex inner products.

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