16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.9

Inner Product Spaces

Mathematics — Linear Algebra · inner-products-quadratic-form

Problem

Let VV be a real or complex vector space with an inner product. Show that the quadratic form determined by the inner product satisfies the parallelogram law ∥α+β∥2+∥α−β∥2=2∥α∥2+2∥β∥2.\|\alpha + \beta\|^2 + \|\alpha - \beta\|^2 = 2\|\alpha\|^2 + 2\|\beta\|^2.
Intuitive explanation

The parallelogram law connects the lengths of a parallelogram's sides and diagonals—here, α+β\alpha + \beta and α−β\alpha - \beta are the diagonals, while α\alpha and β\beta are the sides. The key is to expand both sides using the inner product definition of the norm, ∥γ∥2=⟨γ,γ⟩\|\gamma\|^2 = \langle \gamma, \gamma \rangle, and let the bilinearity and symmetry properties do the work: cross terms will cancel on the left, leaving only the pure norm terms on the right.

Intuitive explanation

The parallelogram law connects the lengths of a parallelogram's sides and diagonals—here, α+β\alpha + \beta and α−β\alpha - \beta are the diagonals, while α\alpha and β\beta are the sides. The key is to expand both sides using the inner product definition of the norm, ∥γ∥2=⟨γ,γ⟩\|\gamma\|^2 = \langle \gamma, \gamma \rangle, and let the bilinearity and symmetry properties do the work: cross terms will cancel on the left, leaving only the pure norm terms on the right.

Related theory

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