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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 Schauder basis ： ウィキペディア英語版 Schauder basis In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927,〔see .〕〔Schauder, Juliusz (1928), "Eine Eigenschaft des Haarschen Orthogonalsystems", ''Mathematische Zeitschrift'' 28: 317–320.〕 although such bases were discussed earlier. For example, the Haar basis was given in 1909, and G. Faber discussed in 1910 a basis for continuous functions on an interval, sometimes called a Faber–Schauder system.〔Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) 19: 104–112. ISSN 0012-0456; http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553〕 == Definitions == Let ''V'' denote a Banach space over the field ''F''. A Schauder basis is a sequence of elements of ''V'' such that for every element there exists a ''unique'' sequence of scalars in ''F'' so that :$v\; =\; \backslash sum\_^\backslash infty\; \backslash alpha\_n\; b\_n,$ where the convergence is understood with respect to the norm topology, ''i.e.'', :$\backslash lim\_\; \backslash left\; \backslash |\; v\; -\; \backslash sum\_^n\; \backslash alpha\_k\; b\_k\; \backslash right\backslash |\_V\; =\; 0.$ Schauder bases can also be defined analogously in a general topological vector space. As opposed to a Hamel basis, the elements of the basis must be ordered since the series may not converge unconditionally. A Schauder basis is said to be normalized when all the basis vectors have norm 1 in the Banach space ''V''. A sequence in ''V'' is a basic sequence if it is a Schauder basis of its closed linear span. Two Schauder bases, in ''V'' and in ''W'', are said to be equivalent if there exist two constants and ''C'' such that for every integer and all sequences of scalars, :$c\; \backslash left\; \backslash |\; \backslash sum\_^N\; \backslash alpha\_k\; b\_k\; \backslash right\backslash |\_V\; \backslash le\; \backslash left\; \backslash |\; \backslash sum\_^N\; \backslash alpha\_k\; c\_k\; \backslash right\; \backslash |\_W\; \backslash le\; C\; \backslash left\; \backslash |\; \backslash sum\_^N\; \backslash alpha\_k\; b\_k\; \backslash right\; \backslash |\_V.$ A family of vectors in ''V'' is total if its linear span (the set of finite linear combinations) is dense in ''V''. If ''V'' is a Hilbert space, an orthogonal basis is a ''total'' subset ''B'' of ''V'' such that elements in ''B'' are nonzero and pairwise orthogonal. Further, when each element in ''B'' has norm 1, then ''B'' is an orthonormal basis of ''V''. 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「Schauder basis」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.