# isometric isomorphism

Let $(X,\parallel {\parallel}_{X})$ and $(Y,\parallel {\parallel}_{Y})$ be normed vector spaces^{}. A surjective linear map $T:X\to Y$ is called an *isometric isomorphism* between $X$ and $Y$ if

$${\parallel Tx\parallel}_{Y}={\parallel x\parallel}_{X},\text{for all}x\in X.$$ |

In this case, $X$ and $Y$ are said to be isometrically isomorphic.

Two isometrically isomorphic normed vector spaces share the same , so they are usually identified with each other.

Title | isometric isomorphism |
---|---|

Canonical name | IsometricIsomorphism |

Date of creation | 2013-03-22 17:34:17 |

Last modified on | 2013-03-22 17:34:17 |

Owner | Gorkem (3644) |

Last modified by | Gorkem (3644) |

Numerical id | 8 |

Author | Gorkem (3644) |

Entry type | Definition |

Classification | msc 46B99 |

Related topic | Isometry |

Defines | isometrically isomorphic |