# proof of generalization of the parallelogram law

Let $g(x,y)=\|x+y\|^{2}-\|x\|^{2}$ and $m(x,y)=\langle x,y\rangle+\langle y,x\rangle$. Then

 $g(x,y)=\|y\|^{2}+m(x,y).$

Hence, taking $x_{1}=x_{4}=x,x_{2}=y,x_{3}=z$ we have:

 $\displaystyle\sum_{i=1}^{3}\|x_{i}+x_{i+1}\|^{2}-\sum_{i=1}^{3}\|x_{i}\|^{2}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{3}g(x_{i},x_{i+1})$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{3}\|x_{i}\|^{2}+\sum_{i=1}^{3}m(x_{i},x_{i+1})$ $\displaystyle=$ $\displaystyle\Big{\|}\sum_{i=1}^{3}x_{i}\Big{\|}^{2}.$
Title proof of generalization of the parallelogram law ProofOfGeneralizationOfTheParallelogramLaw 2013-03-22 16:08:58 2013-03-22 16:08:58 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Proof msc 46C05