# Neumann series in Banach algebras

The Neumann series can be generalized to a Banach algebra^{} $\mathcal{A}$ with identity element^{} $e$.
Thus we have:

If $x\in \mathcal{A}$ is such that $$ then $e-x$ is invertible^{} with inverse^{} given by

$${(e-x)}^{-1}=\sum _{n=0}^{\mathrm{\infty}}{x}^{n}$$ |

and

$$\parallel {(e-x)}^{-1}\parallel \le \frac{1}{1-\parallel x\parallel}$$ |

Title | Neumann series in Banach algebras |
---|---|

Canonical name | NeumannSeriesInBanachAlgebras |

Date of creation | 2013-03-22 17:23:19 |

Last modified on | 2013-03-22 17:23:19 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 4 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46H05 |