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In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). Fréchet spaces are locally convex spaces that are complete with respect to a translation invariant metric. In contrast to Banach spaces, the metric need not arise from a norm. Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the Hahn–Banach theorem, the open mapping theorem, and the Banach–Steinhaus theorem, still hold. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces. == Definitions == Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of semi-norms. A topological vector space ''X'' is a Fréchet space if and only if it satisfies the following three properties: * it is locally convex * its topology can be induced by a translation invariant metric, i.e. a metric ''d'': ''X'' × ''X'' → R such that ''d''(''x'', ''y'') = ''d''(''x''+''a'', ''y''+''a'') for all ''a'',''x'',''y'' in ''X''. This means that a subset ''U'' of ''X'' is open if and only if for every ''u'' in ''U'' there exists an ε > 0 such that is a subset of ''U''. * it is a complete metric space Note that there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology. The alternative and somewhat more practical definition is the following: a topological vector space ''X'' is a Fréchet space if and only if it satisfies the following three properties: * it is a Hausdorff space * its topology may be induced by a countable family of semi-norms ||.||''k'', ''k'' = 0,1,2,... This means that a subset ''U'' of ''X'' is open if and only if for every ''u'' in ''U'' there exists ''K''≥0 and ε>0 such that is a subset of ''U''. * it is complete with respect to the family of semi-norms A sequence (''xn'') in ''X'' converges to ''x'' in the Fréchet space defined by a family of semi-norms if and only if it converges to ''x'' with respect to each of the given semi-norms. 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「Fréchet space」の詳細全文を読む スポンサード リンク
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