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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Neumann series ： ウィキペディア英語版 Neumann series A Neumann series is a mathematical series of the form :$\backslash sum\_^\backslash infty\; T^k$ where ''T'' is an operator. Hence, ''Tk'' is a mathematical notation for ''k'' consecutive operations of the operator ''T''. This generalizes the geometric series. 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 equations. It is also important when studying the spectrum of bounded operators. == Properties == Suppose that ''T'' is a bounded operator on the normed vector space ''X''. If the Neumann series converges in the operator norm, then Id – ''T'' is invertible and its inverse is the series: :$(\backslash mathrm\; -\; T)^\; =\; \backslash sum\_^\backslash infty\; T^k$, where $\backslash mathrm$ is the identity operator in ''X''. To see why, consider the partial sums :$S\_n\; :=\; \backslash sum\_^n\; T^k$. Then we have :$\backslash lim\_(\backslash mathrm-T)S\_n\; =\; \backslash lim\_\backslash left(\backslash sum\_^n\; T^k\; -\; \backslash sum\_^n\; T^\backslash right)\; =\; \backslash lim\_\backslash left(\backslash mathrm\; -\; T^\backslash right)\; =\; \backslash mathrm.$ This result on operators is analogous to geometric series in $\backslash mathbb$, in which we find that: :$\backslash frac1\; =\; 1\; +\; x\; +\; x^2\; +\; \backslash cdots$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Neumann series」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.