# Banach spaces with complemented subspaces

Theorem. [Lindenstrauss-Tzafriri]

Let $V$ be a Banach space^{}, such that for each closed subspace $M$ there exists a closed subspace $N$ such that $M\cap N=0$ and $M+N=V$ (i.e. every closed subspace is complemented). Then $V$ is isomorphic to a Hilbert space^{} (i.e. there exists a Hilbert space structure on $V$ that induces the original topology^{} on $V$ as a Banach space).

Title | Banach spaces with complemented subspaces |
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Canonical name | BanachSpacesWithComplementedSubspaces |

Date of creation | 2013-03-22 16:02:59 |

Last modified on | 2013-03-22 16:02:59 |

Owner | aube (13953) |

Last modified by | aube (13953) |

Numerical id | 13 |

Author | aube (13953) |

Entry type | Theorem |

Classification | msc 46C15 |

Synonym | Lindenstrauss-Tzafriri theorem |

Synonym | Lindenstrauss-Tzafriri complemented subspace theorem |