# polarization identity

Theorem [polarization identity^{}] - Let $X$ be an inner product space^{} over $\mathbb{R}$. The following identity holds for every $x,y\in X$:

$$\u27e8x,y\u27e9=\frac{1}{4}({\parallel x+y\parallel}^{2}-{\parallel x-y\parallel}^{2})$$ |

If $X$ is an inner product space over $\u2102$ instead, the identity becomes

$$\u27e8x,y\u27e9=\frac{1}{4}({\parallel x+y\parallel}^{2}-{\parallel x-y\parallel}^{2})+\frac{1}{4}i({\parallel x+iy\parallel}^{2}-{\parallel x-iy\parallel}^{2})$$ |

Remark - This result shows that the inner product^{} of $X$ is determined by the norm. Moreover, it can be shown that if a normed space $V$ the parallelogram law^{}, the above formulas define an inner product compatible with the norm of $V$.

Title | polarization identity |
---|---|

Canonical name | PolarizationIdentity |

Date of creation | 2013-03-22 17:37:20 |

Last modified on | 2013-03-22 17:37:20 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 4 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46C05 |