Neumann series in Banach algebras

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

If $x\in 𝒜$ 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