proof of parallelogram law

The proof supplied here for the parallelogram law uses the properties of norms and inner products. See the entries about these for more details regarding the following calculations.

Proof.
 $\|x+y\|^{2}+\|x-y\|^{2}$ $=\langle x+y,x+y\rangle+\langle x-y,x-y\rangle$ $=\langle x,x+y\rangle+\langle y,x+y\rangle+\langle x,x-y\rangle-\langle y,x-y\rangle$ $=\overline{\langle x+y,x\rangle}+\overline{\langle x+y,y\rangle}+\overline{% \langle x-y,x\rangle}-\overline{\langle x-y,y\rangle}$ $\displaystyle=\overline{\langle x,x\rangle+\langle y,x\rangle}+\overline{% \langle x,y\rangle+\langle y,y\rangle}+\overline{\langle x,x\rangle-\langle y,% x\rangle}-\left(\overline{\langle x,y\rangle-\langle y,y\rangle}\right)$ $=\overline{\langle x,x\rangle}+\overline{\langle y,x\rangle}+\overline{\langle x% ,y\rangle}+\overline{\langle y,y\rangle}+\overline{\langle x,x\rangle}-% \overline{\langle y,x\rangle}-\overline{\langle x,y\rangle}+\overline{\langle y% ,y\rangle}$ $=\langle x,x\rangle+\langle y,y\rangle+\langle x,x\rangle+\langle y,y\rangle$ $=2\langle x,x\rangle+2\langle y,y\rangle$ $=2\|x\|^{2}+2\|y\|^{2}$.

Title proof of parallelogram law ProofOfParallelogramLaw1 2013-03-22 16:08:15 2013-03-22 16:08:15 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Proof msc 46C05 ProofOfParallelogramLaw AlternateProofOfParallelogramLaw