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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 Infinite-dimensional vector function ： ウィキペディア英語版 Infinite-dimensional vector function Infinite-dimensional vector function refers to a function whose values lie in an infinite-dimensional vector space, such as a Hilbert space or a Banach space. Such functions are applied in most sciences including physics. ==Example== Set $f\_k(t)=t/k^2$ for every positive integer ''k'' and every real number ''t''. Then values of the function :$f(t)=(f\_1(t),f\_2(t),f\_3(t),\backslash ldots)\; \backslash ,$ lie in the infinite-dimensional vector space ''X'' (or $\backslash mathbf\; R^$) of real-valued sequences. For example, :$f(2)\; =\; \backslash left(2,\backslash frac24,\backslash frac29,\backslash frac2,\backslash frac2,\backslash ldots\backslash right).$ As a number of different topologies can be defined on the space ''X'', we cannot talk about the derivative of ''f'' without first defining the topology of ''X'' or the concept of a limit in ''X''. Moreover, for any set ''A'', there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of ''A'' (e.g., the space of functions $A\backslash rightarrow\; K$ with finitely-many nonzero elements, where ''K'' is the desired field of scalars). Furthermore, the argument ''t'' could lie in any set instead of the set of real numbers. 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「Infinite-dimensional vector function」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.