Convergence of Neumann series
Let \(T\) be an operator. A Neumann series is of the following form
\[\sum_{i = 0}^{\infty}T^{i}.\]Theorem: Let \(T\) be a bounded linear operator on a Banach space \(X\). If the Neumann series converges in the operator norm, then \(I-T\) is invertible and its inverse is given by
\[(I - T)^{-1} = \sum_{i = 0}^{\infty}T^{i},\]where \(I\) is the identity operator.
Theorem: