Parallelogram law

A normed vector space $𝑉$ over $𝕂 = ℂ$ or $𝕂 = ℝ$ is induced by a unique inner product iff it satisfies the parallelogram law #m/thm/anal/vec

2 ‖ 𝑥 ‖ 2 + 2 ‖ 𝑦 ‖ 2 = ‖ 𝑥 + 𝑦 ‖ 2 + ‖ 𝑥 - 𝑦 ‖ 2

for all $𝑥, 𝑦 \in 𝑉$ , where the unique inner product is given by the polarization identity

⟨ 𝑦 | 𝑥 ⟩ = ‖ 𝑥 + 𝑦 ‖ 2 - ‖ 𝑥 - 𝑦 ‖ 2 4 + 𝑖 ‖ 𝑖 𝑥 - 𝑦 ‖ 2 - ‖ 𝑖 𝑥 + 𝑦 ‖ 2 4 ⏟_____⏟_____⏟ f o r 𝕂 = ℂ

Proof

#missing/proof

Further properties

The parallelogram law is equivalent to the inequality $2 ‖ 𝑥 ‖ 2 + 2 ‖ 𝑦 ‖ 2 \leq ‖ 𝑥 + 𝑦 ‖ 2 + ‖ 𝑥 - 𝑦 ‖ 2$ by $𝑥 \leftarrow 12 (𝑥 + 𝑦)$ , $𝑦 \leftarrow 12 (𝑥 - 𝑦)$ .

#state/develop | #lang/en | #SemBr