Parallelogram law

If the parallelogram is a rectangle, the two diagonals are of equal lengths AC = BD, so $${\displaystyle 2(A\_B)^{2}+2(B\_C)^{2}=2(A\_C)^{2}}$$ and the statement reduces to the Pythagorean theorem. For the general quadrilateral (with four sides not necessarily equal) Euler's quadrilateral theorem states $${\displaystyle AB^{2}+BC^{2}+CD^{2}+DA^{2}=AC^{2}+BD^{2}+4x^{2},}$$ where ${\displaystyle x}$ is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that ${\displaystyle x=0}$ for a parallelogram, and so the general formula simplifies to the parallelogram law.

Normed vector spaces satisfying the parallelogram law[edit]

Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm for a vector ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})}$ in the real coordinate space ${\displaystyle \mathbb {R} ^{n}}$ is the ${\displaystyle p}$-norm: $${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}$$


Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the ${\displaystyle p}$-norm if and only if ${\displaystyle p=2,}$ the so-called Euclidean norm or standard norm.^[1][2]


For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by: $${\displaystyle \langle x,y\rangle ={\frac {\|x+y\|^{2}-\|x-y\|^{2}}{4}},}$$ or equivalently by $${\displaystyle {\frac {\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}}{2}}\qquad {\text{ or }}\qquad {\frac {\|x\|^{2}+\|y\|^{2}-\|x-y\|^{2}}{2}}.}$$


In the complex case it is given by: $${\displaystyle \langle x,y\rangle ={\frac {\|x+y\|^{2}-\|x-y\|^{2}}{4}}+i{\frac {\|ix-y\|^{2}-\|ix+y\|^{2}}{4}}.}$$


For example, using the ${\displaystyle p}$-norm with ${\displaystyle p=2}$ and real vectors ${\displaystyle x}$ and ${\displaystyle y,}$ the evaluation of the inner product proceeds as follows: $${\displaystyle {\begin{aligned}\langle x,y\rangle &={\frac {\|x+y\|^{2}-\|x-y\|^{2}}{4}}\\[4mu]&={\tfrac {1}{4}}\left(\sum _{i}|x_{i}+y_{i}|^{2}-\sum _{i}|x_{i}-y_{i}|^{2}\right)\\[2mu]&={\tfrac {1}{4}}\left(4\sum _{i}x_{i}y_{i}\right)\\&=x\cdot y,\\\end{aligned}}}$$ which is the standard dot product of two vectors.


Another necessary and sufficient condition for there to exist an inner product that induces the given norm ${\displaystyle \|\cdot \|}$ is for the norm to satisfy Ptolemy's inequality:^[3] $${\displaystyle \|x-y\|\,\|z\|~+~\|y-z\|\,\|x\|~\geq ~\|x-z\|\,\|y\|\qquad {\text{ for all vectors }}x,y,z.}$$