Parallelogram law

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The parallelogram law in elementary geometry

In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. In case the parallelogram is a rectangle, the two diagonals are of equal lengths and the statement reduces to the Pythagorean theorem. But in general, the square of the length of neither diagonal is the sum of the squares of the lengths of two sides.

The parallelogram law in inner product spaces

In inner product spaces, the statement of the parallelogram law reduces to the algebraic identity

<math>2\|x\|^2+2\|y\|^2=\|x+y\|^2+\|x-y\|^2<math>

where

<math>\|x\|^2=\langle x, x\rangle.<math>

Which normed vector spaces satisfy the parallelogram law?

Most normed vector spaces do not have inner products, but all normed vector spaces have norms (hence the name), and thus one can evaluate the expressions on both sides of "=" in the identity above. A remarkable fact is that the identity above holds only if the norm is one that arises in the usual way from an inner product, because, if the identity above holds, then the function

<math>\langle x, y\rangle={\|x+y\|^2-\|x\|^2-\|y\|^2\over 2}<math>

is an inner product whose norm is precisely this one.

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