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The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments. ==Formulation== Let ''E'' be a real vector space, ''F'' ⊂ ''E'' a vector subspace, and let ''K'' ⊂ ''E'' be a convex cone. A linear functional ''φ'': ''F'' → R is called ''K''-''positive'', if it takes only non-negative values on the cone ''K'': : A linear functional ''ψ'': ''E'' → R is called a ''K''-positive ''extension'' of ''φ'', if it is identical to ''φ'' in the domain of ''φ'', and also returns a value of at least 0 for all points in the cone ''K'': : In general, a ''K''-positive linear functional on ''F'' can not be extended to a -positive linear functional on ''E''. Already in two dimensions one obtains a counterexample taking ''K'' to be the upper halfplane with the open negative ''x''-axis removed. If ''F'' is the real axis, then the positive functional ''φ''(''x'', 0) = ''x'' can not be extended to a positive functional on the plane. However, the extension exists under the additional assumption that for every ''y'' ∈ ''E'' there exists ''x''∈''F'' such that ''y'' − ''x'' ∈''K''; in other words, if ''E'' = ''K'' + ''F''. 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「M. Riesz extension theorem」の詳細全文を読む スポンサード リンク
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