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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 Linear form ： ウィキペディア英語版 Linear form In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars. In ℝ''n'', if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the matrix product with the row vector on the left and the column vector on the right.  In general, if ''V'' is a vector space over a field ''k'', then a linear functional ''f'' is a function from ''V'' to ''k'' that is linear: :$f(\backslash vec+\backslash vec)\; =\; f(\backslash vec)+f(\backslash vec)$ for all $\backslash vec,\; \backslash vec\backslash in\; V$ :$f(a\backslash vec)\; =\; af(\backslash vec)$ for all $\backslash vec\backslash in\; V,\; a\backslash in\; k.$ The set of all linear functionals from ''V'' to ''k'', Hom''k''(''V'',''k''), forms a vector space over ''k'' with the addition of the operations of addition and scalar multiplication (defined pointwise).  This space is called the dual space of ''V'', or sometimes the algebraic dual space, to distinguish it from the continuous dual space.  It is often written ''V''∗ or ''V′'' when the field ''k'' is understood. == Continuous linear functionals == If ''V'' is a topological vector space, the space of continuous linear functionals — the ''continuous dual'' — is often simply called the dual space.  If ''V'' is a Banach space, then so is its (continuous) dual.  To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the ''algebraic dual''.  In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, although this is not true in infinite dimensions. 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「Linear form」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.