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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

1.1

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

A1A2algebra

2.1

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.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, p.39 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 15 (standard reference, not scraped)