Neumann Series Theorem
Theorem
Let $X$ be a Banach space.
Let $\map {CL} X$ be the continous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $A \in \map {CL} X$ be such that $\norm A < 1$.
Let $\circ$ be the composition of mappings.
Let $I$ be the identity mapping.
Then:
- $I - A$ is invertible in $\map {CL} X$
- $\ds \paren {I - A}^{-1} = \sum_{n \mathop = 0}^\infty A^n$
- $\norm {\paren{I - A}^{-1} } \le \dfrac 1 {1 - \norm A}$
Corollary 1
The mapping $I - A : X \to X$ is bijective.
Corollary 2
The mapping $\paren {I - A}^{-1} : X \to X$ is continuous.
Proof
Let $\ds S_k := \sum_{n \mathop = 0}^k A^n$.
$\ds \sum_{n \mathop = 0}^\infty A^n$ converges absolutely
By Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative:
- $\forall n \in \N : \norm {A^n} \le \norm A^n$
By assumption, $\norm A < 1$.
By Sum of Infinite Geometric Sequence, $\ds \sum_{n \mathop = 0}^\infty \norm A^n$ converges.
By series comparison, $\ds \sum_{n \mathop = 0}^\infty \norm {A^n}$ converges too.
By definition, $\ds \sum_{n \mathop = 0}^\infty A^n$ is absolutely convergent.
$\Box$
$\ds \sum_{n \mathop = 0}^\infty A^n$ converges
By assumption, $X$ is Banach.
By Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space, $\map {CL} X$ is Banach too.
Let $\ds S := \sum_{n \mathop = 0}^\infty A^n$.
We have that Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach.
Then $S$ converges in $\map {CL} X$.
In other words:
- $\ds \lim_{k \mathop \to \infty} S_k = S$
$\Box$
Inverse of $\paren {I - A}$ is $S$
We have that:
| \(\ds A S_k\) | \(=\) | \(\ds A \circ \paren{\sum_{n \mathop = 0}^k A^n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^k A^{n + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\sum_{n \mathop = 0}^k A^n} \circ A\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds S_k A\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n' \mathop = 1}^{k + 1} A^{n'}\) | $n' = n + 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds S_{k + 1} - I\) |
Furthermore:
| \(\ds \norm {A S_k - A S}\) | \(=\) | \(\ds \norm {A \paren {S_k - S} }\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \norm {A} \norm {S_k - S}\) | Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \norm {S_k - S}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \lim_{k \mathop \to \infty} \norm {S_k A - S A}\) | \(\le\) | \(\ds \lim_{k \mathop \to \infty} \norm {S_k - S}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \norm {\lim_{k \mathop \to \infty} \paren {S_k - S} }\) | Supremum Norm is Norm, Norm is Continuous, Limit of Composite Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
| \(\ds \norm {S_k A - S A}\) | \(=\) | \(\ds \norm {\paren {S_k - S} A}\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \norm {A} \norm {S_k - S}\) | Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \norm {S_k - S}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \lim_{k \mathop \to \infty} \norm {S_k A - S A}\) | \(\le\) | \(\ds \lim_{k \mathop \to \infty} \norm {S_k - S}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \norm {\lim_{k \mathop \to \infty} \paren {S_k - S} }\) | Supremum Norm is Norm, Norm is Continuous, Limit of Composite Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
Hence:
| \(\ds \lim_{k \mathop \to \infty} S_k A\) | \(=\) | \(\ds S A\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{k \mathop \to \infty} A S_k\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds A \lim_{k \mathop \to \infty} S_k\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds A S\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{k \mathop \to \infty} \paren {S_{k+1} - I}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds S - I\) |
That is:
- $S A = A S = S - I$
It follows that:
| \(\ds I\) | \(=\) | \(\ds S - S A\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds S \paren {I - A}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds S - A S\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {I - A} S\) |
Therefore, $I - A$ is invertible in $\map {CL} X$.
Furthermore:
- $\ds \paren {I - A}^{-1} = S = \sum_{k \mathop = 0}^\infty A^k$
Moreover:
| \(\ds \norm {\paren {I - A}^{-1} }\) | \(=\) | \(\ds \norm {\sum_{k \mathop = 0}^\infty A^k}\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{k \mathop = 0}^\infty \norm {A^k}\) | Norm Axiom $\text N 3$: Triangle Inequality | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{k \mathop = 0}^\infty \norm A^k\) | Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \frac 1 {1 - \norm A}\) | Sum of Infinite Geometric Sequence |
$\blacksquare$
Also see
Source of Name
This entry was named for Carl Gottfried Neumann.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations