Suppose that ${\displaystyle T}$  is a bounded linear operator on the normed vector space ${\displaystyle X}$ . If the Neumann series converges in the operator norm, then ${\displaystyle I-T}$  is invertible and its inverse is the series:

${\displaystyle (I-T)^{-1}=\sum _{k=0}^{\infty }T^{k}}$ ,

where ${\displaystyle I}$  is the identity operator in ${\displaystyle X}$ . To see why, consider the partial sums

${\displaystyle S_{n}:=\sum _{k=0}^{n}T^{k}}$ .

Then we have

${\displaystyle \lim _{n\rightarrow \infty }(I-T)S_{n}=\lim _{n\rightarrow \infty }\left(\sum _{k=0}^{n}T^{k}-\sum _{k=0}^{n}T^{k+1}\right)=\lim _{n\rightarrow \infty }\left(I-T^{n+1}\right)=I}$ 

This result on operators is analogous to geometric series in ${\displaystyle \mathbb {R} }$ .

One case in which convergence is guaranteed is when ${\displaystyle X}$  is a Banach space and ${\displaystyle |T|<1}$  in the operator norm; another compatible case is that ${\textstyle \sum _{k=0}^{\infty }|T^{k}|}$  converges. However, there are also results which give weaker conditions under which the series converges.

Let ${\displaystyle C\in \mathbb {R} ^{3\times 3}}$  be given by:

${\displaystyle {\begin{pmatrix}0&{\frac {1}{2}}&{\frac {1}{4}}\\{\frac {5}{7}}&0&{\frac {1}{7}}\\{\frac {3}{10}}&{\frac {3}{5}}&0\end{pmatrix}}.}$ 

For the Neumann series ${\textstyle \sum _{k=0}^{n}C^{k}}$  to converge to ${\displaystyle (I-C)^{-1}}$  as ${\displaystyle n}$  goes to infinity, the matrix norm of ${\displaystyle C}$  must be smaller than unity. This norm is

${\displaystyle {\begin{aligned}||C||_{\infty }&=\max _{i}\sum _{j}|c_{ij}|=\max \left\lbrace {\frac {3}{4}},{\frac {6}{7}},{\frac {9}{10}}\right\rbrace ={\frac {9}{10}}<1,\end{aligned}}}$ 

confirming that the Neumann series converges.

A truncated Neumann series can be used for approximate matrix inversion. To approximate the inverse of an invertible matrix ${\displaystyle A}$ , consider that

${\displaystyle {\begin{aligned}A^{-1}&=(I-I+A)^{-1}\\&=(I-(I-A))^{-1}\\&=(I-T)^{-1}\end{aligned}}}$ 

for ${\displaystyle T=(I-A).}$  Then, using the Neumann series identity that ${\textstyle \sum _{k=0}^{\infty }T^{k}=(1-T)^{-1}}$  if the appropriate norm condition on ${\displaystyle T=(I-A)}$  is satisfied, ${\textstyle A^{-1}=(I-(I-A))^{-1}=\sum _{k=0}^{\infty }(I-A)^{k}.}$  Since these terms shrink with increasing ${\displaystyle k,}$  given the conditions on the norm, then truncating the series at some finite ${\displaystyle n}$  may give a practical approximation to the inverse matrix:

${\displaystyle A^{-1}\approx \sum _{k=0}^{n}(I-A)^{k}.}$ 

A corollary is that the set of invertible operators between two Banach spaces ${\displaystyle B}$  and ${\displaystyle B'}$  is open in the topology induced by the operator norm. Indeed, let ${\displaystyle S:B\to B'}$  be an invertible operator and let ${\displaystyle T:B\to B'}$  be another operator. If ${\displaystyle |S-T|<|S^{-1}|^{-1}}$ , then ${\displaystyle T}$  is also invertible. Since ${\displaystyle |I-S^{-1}T|<1}$ , the Neumann series ${\textstyle \sum _{k=0}^{\infty }(I-S^{-1}T)^{k}}$  is convergent. Therefore, we have

${\displaystyle T^{-1}S=(I-(I-S^{-1}T))^{-1}=\sum _{k=0}^{\infty }(I-S^{-1}T)^{k}}$ 

Taking the norms, we get

${\displaystyle |T^{-1}S|\leq {\frac {1}{1-|I-(S^{-1}T)|}}}$ 

The norm of ${\displaystyle T^{-1}}$  can be bounded by

${\displaystyle |T^{-1}|\leq {\tfrac {1}{1-q}}|S^{-1}|\quad {\text{where}}\quad q=|S-T|\,|S^{-1}|.}$ 

The Neumann series has been used for linear data detection in massive multiuser multiple-input multiple-output (MIMO) wireless systems. Using a truncated Neumann series avoids computation of an explicit matrix inverse, which reduces the complexity of linear data detection from cubic to square.^[1]

Another application is the theory of propagation graphs which takes advantage of Neumann series to derive closed form expressions for transfer functions.

• Werner, Dirk (2005). Funktionalanalysis (in German). Springer Verlag. ISBN 3-540-43586-7.