Exercise 2.2 (Polarization Identity)

Lemma 2.1 (Polarization Identity). Suppose ⟨⋅,⋅⟩ is an inner product on a vector space V . Then foral all v , w ∈ V ,

⟨ v , w ⟩ = 1 4 ( ⟨ v + w , v + w ⟩ − ⟨ v − w , v − w ⟩ ) .

Exercise 2.2. Prove the preceding lemma.

Answers

Proof. Using bilinearity several times and applying symmetry in the last step, we obtain

⟨ v + w , v + w ⟩ − ⟨ v − w , v − w ⟩ = ⟨ v , v + w ⟩ + ⟨ w , v + w ⟩ − ⟨ v , v − w ⟩ + ⟨ w , v − w ⟩ = ⟨ v , v ⟩ + ⟨ v , w ⟩ + ⟨ w , v ⟩ + ⟨ w , w ⟩ − ⟨ v , v ⟩ + ⟨ v , w ⟩ + ⟨ w , v ⟩ − ⟨ w , w ⟩ = 2 ⟨ v , w ⟩ + 2 ⟨ w , v ⟩ = 4 ⟨ v , w ⟩ □
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2023-10-30 10:59
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