Parallelogram law

The parallelogram law states that the sum of the squared norms of two vectors’ sum and difference equals twice the sum of their squared norms.

Inner product: An operation that assigns a scalar to two vectors and captures notions such as length and angle. Expanding the squared norms with an inner product makes the identity immediate.
Hilbert space: A complete inner-product space with a norm induced by its inner product. Every Hilbert-space norm obeys the parallelogram law.
Vector space: A set with addition and scalar multiplication satisfying the vector-space axioms. The law compares sums and differences formed within a vector space.
Taxicab norm: The vector norm that sums the absolute values of coordinates. In dimension two, its unit ball is a diamond and it does not satisfy the law.
Norm: A function that assigns a nonnegative length or size to vectors and satisfies specified axioms. The law is a condition on a norm, not on vector addition alone.
Banach space: A complete normed vector space. The law distinguishes inner-product norms from many other Banach-space norms.
Vector addition: The operation that combines two vectors into their sum. Both diagonal vectors in the law are formed by adding or subtracting its inputs.
Maximum norm: The vector norm given by the largest absolute coordinate. Its squared lengths generally violate the identity, unlike Euclidean lengths.
Polarization identity: An identity that recovers an inner product from the norms of vector sums and differences. It reverses the law’s relationship by reconstructing inner products from norms.
Lp space: A space of functions or sequences whose absolute values have a finite p-th power integral or sum. For p other than two, its usual norm generally fails the parallelogram law.

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