# Schauder lemma

The following theorem is in the functional analysis^{} literature generally referred to as the *Schauder lemma*. It is a version of the open mapping theorem^{} in Fréchet spaces and is often used to verify the open-ness of linear, continuous maps^{}.

Theorem. Let $E,F$ be Fréchet spaces. Denote by ${\mathcal{U}}_{0}(E),{\mathcal{U}}_{0}(F)$ the zero neighborhood filter of $E$ and $F$ respectively. Let $T:E\to F$ be a linear and continuous map which is *almost open*, i.e.

${\forall}_{U\in {\mathcal{U}}_{0}(E)}{\overline{T(U)}}^{F}$ | $\in {\mathcal{U}}_{0}(F)$ |

Then $T$ is open.

Title | Schauder lemma |
---|---|

Canonical name | SchauderLemma |

Date of creation | 2013-03-22 19:09:44 |

Last modified on | 2013-03-22 19:09:44 |

Owner | karstenb (16623) |

Last modified by | karstenb (16623) |

Numerical id | 4 |

Author | karstenb (16623) |

Entry type | Theorem |

Classification | msc 54E50 |

Classification | msc 46A30 |

Related topic | OpenMappingTheorem |

Defines | almost open set |