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 ${\displaystyle {\text{Id}}-T}$  is invertible and its inverse is the series:

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

where ${\displaystyle \mathrm {Id} }$  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

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

This result on operators is analogous to geometric series in ${\displaystyle \mathbb {R} }$ , in which we find that:

${\displaystyle (1-x)\cdot (1+x+x^{2}+\cdots +x^{n-1}+x^{n})=1-x^{n+1},}$ 

${\displaystyle 1+x+x^{2}+\cdots ={\frac {1}{1-x}}.}$ 

One case in which convergence is guaranteed is when ${\displaystyle X}$  is a Banach space and ${\displaystyle |T|<1}$  in the operator norm or ${\displaystyle \sum |T^{n}|}$  is convergent. 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:

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

We need to show that C is smaller than unity in some norm. Therefore, we calculate:

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

Thus, we know from the statement above that ${\displaystyle (I-C)^{-1}}$  exists.

A truncated Neumann series can be used for approximate matrix inversion. To approximate the inverse of an invertible matrix ${\displaystyle \mathbf {A} }$ , we can assign the linear operator as:

${\displaystyle T(\mathbf {x} )=(\mathbf {I} -\mathbf {A} )\mathbf {x} }$ 

where ${\displaystyle \mathbf {I} }$  is the identity matrix. If the norm condition on ${\displaystyle T}$  is satisfied, then truncating the series at ${\displaystyle n}$ , we get:

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

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 ${\displaystyle |\mathrm {Id} -S^{-1}T|<1}$ , the Neumann series ${\displaystyle \sum (\mathrm {Id} -S^{-1}T)^{k}}$  is convergent. Therefore, we have

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

Taking the norms, we get

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

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 expression for the transfer function.

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