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In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition of continuous linear mappings stop being jointly continuous at the level of Banach spaces, where , a locally convex vector space. is its dual of continuous linear functionals equipped with any locally convex topology such that the evaluation mapping is separately continuous. If the evaluation is assumed to be jointly continuous, then there are neighborhoods and of zero such that . However, this means that is contained in the polar of the open set ; so it is bounded in . Thus admits a bounded neighborhood of zero, and is thus a normed vector space.}} for any compatible topology on the spaces of continuous linear mappings. Mappings between convenient vector spaces are smooth or if they map smooth curves to smooth curves. This leads to a Cartesian closed category of smooth mappings between -open subsets of convenient vector spaces (see property 6 below). The corresponding calculus of smooth mappings is called ''convenient calculus''. It is weaker than any other reasonable notion of differentiability, it is easy to apply, but there are smooth mappings which are not continuous (see Note 1). This type of calculus alone is not useful in solving equations. ==The -topology== Let be a locally convex vector space. A curve is called ''smooth'' or if all derivatives exist and are continuous. Let be the space of smooth curves. It can be shown that the set of smooth curves does not depend entirely on the locally convex topology of ''E'', only on its associated bornology (system of bounded sets); see (), 2.11. The final topologies with respect to the following sets of mappings into coincide; see (), 2.13. * . * The set of all Lipschitz curves (so that } is bounded in , for each ). * The set of injections where runs through all bounded absolutely convex subsets in , and where is the linear span of equipped with the Minkowski functional . * The set of all Mackey-convergent sequences (there exists a sequence with bounded). This topology is called the -''topology'' on and we write for the resulting topological space. In general (on the space of smooth functions with compact support on the real line, for example) it is finer than the given locally convex topology, it is not a vector space topology, since addition is no longer jointly continuous. Namely, even . The finest among all locally convex topologies on which are coarser than is the bornologification of the given locally convex topology. If is a Fréchet space, then . ==Convenient vector spaces== A locally convex vector space is said to be a ''convenient vector space'' if one of the following equivalent conditions holds (called -completeness); see (), 2.14. * For any the (Riemann-) integral exists in . * Any Lipschitz curve in is locally Riemann integrable. * Any ''scalar wise'' curve is : A curve is smooth if and only if the composition is in for all , where is the dual of all continuous linear functionals on . * * Equivalently, for all , the dual of all bounded linear functionals. * * Equivalently, for all , where is a subset of which recognizes bounded subsets in . * Any Mackey-Cauchy-sequence (i. e., for some in ) converges in . This is visibly a mild completeness requirement. * If is bounded closed absolutely convex, then is a Banach space. * If is scalar wise , then is , for . * If is scalarwise then is differentiable at 0. Here a mapping is called if all derivatives up to order exist and are Lipschitz, locally on . ==Smooth mappings== Let and be convenient vector spaces, and let be -open. A mapping is called ''smooth'' or , if the composition for all . See(), 3.11. 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「Convenient vector space」の詳細全文を読む スポンサード リンク
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