# yet another proof of parallelogram law

Define $g(\u03f5)=\u27e8x+\u03f5y,x+\u03f5y\u27e9$, where $\u03f5$ is real. Then $g(\u03f5)=\u27e8x,x\u27e9+\u03f5(\u27e8y,x\u27e9+\u27e8x,y\u27e9)+{\u03f5}^{2}\u27e8y,y\u27e9.$ Hence,

$${\parallel x+y\parallel}^{2}+{\parallel x-y\parallel}^{2}=g(1)+g(-1)=2\u27e8x,x\u27e9+2\u27e8y,y\u27e9=2{\parallel x\parallel}^{2}+2{\parallel y\parallel}^{2}.$$ |

Title | yet another proof of parallelogram law^{} |
---|---|

Canonical name | YetAnotherProofOfParallelogramLaw |

Date of creation | 2013-03-22 16:08:23 |

Last modified on | 2013-03-22 16:08:23 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 5 |

Author | Mathprof (13753) |

Entry type | Proof |

Classification | msc 46C05 |