Inner product space

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 vec

2‖𝑥‖2+2‖𝑦‖2=‖𝑥+𝑦‖2+‖𝑥−𝑦‖2

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

⟨𝑦|𝑥⟩=‖𝑥+𝑦‖2−‖𝑥−𝑦‖24+𝑖‖𝑖𝑥−𝑦‖2−‖𝑖𝑥+𝑦‖24⏟____⏟____⏟for 𝕂=ℂ

Proof

proof

Further properties

1. The parallelogram law is equivalent to the inequality 2‖𝑥‖2 +2‖𝑦‖2 ≤‖𝑥 +𝑦‖2 +‖𝑥 −𝑦‖2 by 𝑥 ←12(𝑥 +𝑦), 𝑦 ←12(𝑥 −𝑦).

Corollaries

• Lebesgue space forms an inner product space iff p=2

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