翻訳と辞書 Words near each other ・ Hahnenkopf ・ Hahnenmoos Pass ・ Hahnensee ・ Hahnerberger Siepen ・ Hahnfjella ・ Hahnheim ・ Hahnia ・ Hahnodon ・ Hahnsberg ・ Hahnstätten ・ Hahnstätten (Verbandsgemeinde) ・ Hahntennjoch ・ Hahnville High School ・ Hahnville, Louisiana ・ Hahnweiler ・ Hahn–Banach theorem ・ Hahn–Exton q-Bessel function ・ Hahn–Kolmogorov theorem ・ Haho of Maui ・ Haho Prefecture ・ Haho, Burkina Faso ・ Hahoe byeolsingut talnori ・ Hahoe Folk Village ・ Hahoetal ・ Hahon language ・ HaHotrim ・ HAHS ・ Hahót ・ HAI ・ Hai (film) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hahn–Banach theorem ： ウィキペディア英語版 Hahn–Banach theorem In mathematics, the Hahn–Banach Theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of Hahn–Banach theorem is known as Hahn–Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry. It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case〔for the space of continuous functions on an interval〕 was proved earlier (in 1912) by Eduard Helly, and a general extension theorem from which the Hahn–Banach theorem can be derived was proved in 1923 by Marcel Riesz.〔See M. Riesz extension theorem. According to , the argument was known to Riesz already in 1918.〕 == Formulation == The most general formulation of the theorem needs some preparation. Given a real vector space , a function is called sublinear if * Positive Homogeneity: for all , * Subadditivity: for all . Every seminorm on (in particular, every norm on ) is sublinear. Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets. Hahn–Banach Theorem . If is a sublinear function, and is a linear functional on a linear subspace which is dominated by on , i.e. :$\backslash varphi(x)\; \backslash leq\; p(x)\backslash qquad\backslash forall\; x\; \backslash in\; U$ then there exists a linear extension of to the whole space , i.e., there exists a linear functional such that :$\backslash psi(x)=\backslash varphi(x)\backslash qquad\backslash forall\; x\backslash in\; U,$ :$\backslash psi(x)\; \backslash le\; p(x)\backslash qquad\backslash forall\; x\backslash in\; V.$ Hahn–Banach Theorem (Alternate Version). Set or and let be a -vector space with a seminorm . If is a -linear functional on a -linear subspace of which is dominated by on in absolute value, :$|\backslash varphi(x)|\backslash leq\; p(x)\backslash qquad\backslash forall\; x\; \backslash in\; U$ then there exists a linear extension of to the whole space , i.e., there exists a -linear functional such that :$\backslash psi(x)=\backslash varphi(x)\backslash qquad\backslash forall\; x\backslash in\; U,$ :$|\backslash psi(x)|\; \backslash le\; p(x)\backslash qquad\backslash forall\; x\backslash in\; V.$ In the complex case of the alternate version, the -linearity assumptions demand, in addition to the assumptions for the real case, that for every vector , we have and . The extension is in general not uniquely specified by and the proof gives no explicit method as to how to find . The usual proof for the case of an infinite dimensional space uses Zorn's lemma or, equivalently, the axiom of choice. It is now known (see section 4.0) that the ultrafilter lemma, which is slightly weaker than the axiom of choice, is actually strong enough. It is possible to relax slightly the subadditivity condition on , requiring only that (Reed and Simon, 1980): :$p(ax+by)\backslash leq|a|\; \backslash ,\; p(x)\; +\; |b|\; \backslash ,\; p(y),\backslash qquad\; x,y\backslash in\; V,\backslash quad\; |a|+|b|\backslash leq1.$ This reveals the intimate connection between the Hahn–Banach theorem and convexity. The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the (HAHNBAN file ). 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「Hahn–Banach theorem」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.