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In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on ''V'' together with a naturally induced linear structure. Dual vector spaces for finite-dimensional vector spaces show up in tensor analysis. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. There are two types of dual spaces: the ''algebraic dual space'', and the ''continuous dual space''. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space. == Algebraic dual space == Given any vector space ''V'' over a field ''F'', the dual space ''V''∗ is defined as the set of all linear maps (linear functionals). The dual space ''V''∗ itself becomes a vector space over ''F'' when equipped with an addition and scalar multiplication satisfying: : for all ''φ'' and , , and . Elements of the algebraic dual space ''V''∗ are sometimes called covectors or one-forms. The pairing of a functional ''φ'' in the dual space ''V''∗ and an element ''x'' of ''V'' is sometimes denoted by a bracket: or . The pairing defines a nondegenerate bilinear mapping〔In many areas, such as quantum mechanics, is reserved for a sesquilinear form defined on .〕 . 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「dual space」の詳細全文を読む スポンサード リンク
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