 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2007/b/5/a/solution

Expand both squared norms using sesquilinearity and conjugate symmetry. Their mixed terms

⟨ u, v ⟩ + ⟨ v, u ⟩

appear with opposite signs and cancel. Hence

∥ u + v ∥^{2} + ∥ u - v ∥^{2} = 2∥ u ∥^{2} + 2∥ v ∥^{2},

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