TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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The parallelogram law

Statement

In every real or complex inner-product space, for all vectors $x, y$ ,

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

Facts & Assumptions

[A1]

The pairing is linear in the first argument, conjugate-linear in the second and conjugate symmetric, and $∥ v ∥^{2} = ⟨ v, v ⟩$ (Real and complex inner-product spaces and their induced length).

[A2]

The induced length is the unique nonnegative square root of the diagonal pairing, so $∥ v ∥^{2} = ⟨ v, v ⟩$ (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product).

Proof

technique · direct

Given: Vectors $x, y$ in a real or complex inner-product space.

1.1A1A2algebra

Expanding with linearity in the first argument, conjugate-linearity in the second and conjugate symmetry gives $∥ x + y ∥^{2} = ⟨ x + y, x + y ⟩ = ∥ x ∥^{2} + ⟨ x, y ⟩ + \overline{⟨ x, y ⟩} + ∥ y ∥^{2}$ and, replacing $y$ by $- y$ , $∥ x - y ∥^{2} = ∥ x ∥^{2} - ⟨ x, y ⟩ - \overline{⟨ x, y ⟩} + ∥ y ∥^{2}$ .

2.1step 1.1algebra∎

The cross terms in step 1.1 cancel when the two expansions are added, so $∥ x + y ∥^{2} + ∥ x - y ∥^{2} = 2 ∥ x ∥^{2} + 2 ∥ y ∥^{2}$ , the parallelogram law.

The parallelogram law

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources