The Parallelogram Law: Which Norms Come from an Inner Product
An interactive live film: two columns compare the squared diagonals of a parallelogram with its squared sides; under the ℓ₂ norm they always match, under ℓ₁ and ℓ∞ they come apart, so those norms come from no inner product.
Narration
Two arrows on the floor, the red u and the blue v, make a parallelogram. Its two diagonals are the purple u plus v and the orange u minus v.
With the usual length, the squared diagonals add up to the squares of all four sides: two times u squared plus two times v squared. This is the parallelogram law.
The rest of the narration is in the book's Companion.
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