# nuclear space

If $E$ is a FrÃÂ©chet space and $({p}_{j})$ an increasing sequence of semi-norms on $E$ defining the topology of $E$, we have

$$E=\underset{\u27f5}{lim}{\widehat{E}}_{{p}_{j}},$$ |

where ${\widehat{E}}_{{p}_{j}}$ is the Hausdorff completion of $(E,{p}_{j})$ and ${\widehat{E}}_{{p}_{j+1}}\to {\widehat{E}}_{{p}_{j}}$ the canonical morphism. Here ${\widehat{E}}_{{p}_{j}}$ is a Banach space^{} for the induced norm^{} ${\widehat{p}}_{j}$.

A FrÃÂ©chet space $E$ is said to be *nuclear* if the topology of $E$ can be defined by an increasing sequence of semi-norms ${p}_{j}$ such that each canonical morphism ${\widehat{E}}_{{p}_{j+1}}\to {\widehat{E}}_{{p}_{j}}$ of Banach spaces is nuclear.

Recall that a morphism $f:E\to F$ of complete^{} locally convex spaces is said to be nuclear if $f$ can be written as

$$f(x)=\sum {\lambda}_{j}{\xi}_{j}(x){y}_{j}$$ |

where $({\lambda}_{j})$ is a sequence of scalars with $$,${\xi}_{j}\in {E}^{\prime}$ an equicontinuous sequence of linear forms and ${y}_{j}\in F$ a bounded^{} sequence.

Title | nuclear space |
---|---|

Canonical name | NuclearSpace |

Date of creation | 2013-03-22 16:37:28 |

Last modified on | 2013-03-22 16:37:28 |

Owner | Simone (5904) |

Last modified by | Simone (5904) |

Numerical id | 6 |

Author | Simone (5904) |

Entry type | Definition |

Classification | msc 46B20 |