# Pythagorean theorem in inner product spaces

- Let $X$ be an inner product space^{} (over $\mathbb{R}$ or $\u2102$) and $x,y\in X$ two orthogonal vectors^{}. Then

$${\parallel x+y\parallel}^{2}={\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}.$$ |

Proof : As $x\u27c2y$ one has $\u27e8x,y\u27e9=0$. Then

${\parallel x+y\parallel}^{2}$ | $=$ | $\u27e8x+y,x+y\u27e9$ | ||

$=$ | $\u27e8x,x\u27e9+\u27e8x,y\u27e9+\u27e8y,x\u27e9+\u27e8y,y\u27e9$ | |||

$=$ | ${\parallel x\parallel}^{2}+\u27e8x,y\u27e9+\overline{\u27e8x,y\u27e9}+{\parallel y\parallel}^{2}$ | |||

$=$ | ${\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\square}$ |

$Remark-$ This theorem is valid (with the same proof) for spaces with a semi-definite inner product^{}.

Title | Pythagorean theorem in inner product spaces |
---|---|

Canonical name | PythagoreanTheoremInInnerProductSpaces |

Date of creation | 2013-03-22 17:32:13 |

Last modified on | 2013-03-22 17:32:13 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46C05 |

Synonym | Pythagoras theorem in inner product spaces |

Related topic | PythagorasTheorem |