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| A '''Neumann series''' is a [[series (mathematics)|mathematical series]] of the form
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| :<math> \sum_{k=0}^\infty T^k </math>
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| where ''T'' is an [[Operator (mathematics)|operator]]. Hence, ''T<sup>k</sup>'' is a mathematical notation for ''k'' consecutive operations of the operator ''T''. This generalizes the [[geometric series]].
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| The series is named after the mathematician [[Carl Neumann]], who used it in 1877 in the context of [[potential theory]]. The Neumann series is used in [[functional analysis]]. It forms the basis of the [[Liouville-Neumann series]], which is used to solve [[Fredholm integral equation]]s. It is also important when studying the [[spectrum (functional analysis)|spectrum]] of bounded operators.
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| == Properties ==
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| Suppose that ''T'' is a bounded operator on the [[normed vector space]] ''X''. If the Neumann series [[Convergent series|converges]] in the [[operator norm]], then Id – ''T'' is [[Invertible matrix|invertible]] and its inverse is the series:
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| :<math> (\mathrm{Id} - T)^{-1} = \sum_{k=0}^\infty T^k </math>,
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| where <math> \mathrm{Id} </math> is the [[identity operator]] in ''X''. To see why, consider the partial sums
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| :<math>S_n := \sum_{k=0}^n T^k</math>.
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| Then we have
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| :<math>\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}.</math> | |
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| One case in which convergence is guaranteed is when ''X'' is a [[Banach space]] and |''T''| < 1 in the operator norm. However, there are also results which give weaker conditions under which the series converges.
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| == The set of invertible operators is open ==
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| A corollary is that the set of invertible operators between two Banach spaces ''B'' and ''B''' is open in the topology induced by the operator norm. Indeed, let ''S'' : ''B'' → ''B''<nowiki>'</nowiki> be an invertible operator and let ''T'': ''B'' → ''B''<nowiki>'</nowiki> be another operator. If |''S'' – ''T'' | < |''S''<sup>–1</sup>|<sup>–1</sup>, then ''T'' is also invertible. This follows by writing ''T'' as
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| :<math> T = S ( \mathrm{Id} - (\mathrm{Id} - S^{-1} T ))\, </math>
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| and applying the result in the previous section on the second factor. The norm of ''T''<sup>–1</sup> can be bounded by
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| :<math> |T^{-1}| \le \tfrac{1}{1-q} |S^{-1}| \quad\text{where}\quad q = |S-T| \, |S^{-1}|. </math>
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| == References ==
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| * {{cite book| last=Werner| first=Dirk| year=2005| title=Funktionalanalysis | language=German| publisher=Springer Verlag| isbn=3-540-43586-7}}
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| {{iw-ref|de|Neumann-Reihe}}
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| [[Category:Functional analysis]]
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| [[Category:Mathematical series]]
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