 Solution

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

Expand both squared norms using the inner product:

∥ x \pm y ∥^{2} = ∥ x ∥^{2} + ∥ y ∥^{2} \pm 2 Re ⟨ x, y ⟩ .

Adding cancels the cross terms, giving the parallelogram law

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

The real part may be omitted in a real inner-product space.

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