# 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 $\|x\|<1$ then $e-x$ is invertible with inverse given by

 $(e-x)^{-1}=\sum_{n=0}^{\infty}x^{n}$

and

 $\|(e-x)^{-1}\|\leq\frac{1}{1-\|x\|}$
Title Neumann series in Banach algebras NeumannSeriesInBanachAlgebras 2013-03-22 17:23:19 2013-03-22 17:23:19 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 46H05