 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ib/paper-3/1a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Expand using the bilinearity and symmetry of the real inner product:

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

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

For the maximum norm on

R^{2}

, take

x = (1, 0)

and

y = (0, 1)

x, y, x + y, x - y

have maximum norm one. The left side of the parallelogram law is then

2

, whereas the right side is

4

. The maximum norm cannot come from an inner product.

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