Кіріспе
Математикада, атап айтқанда, нақты талдау және функционалдық талдау салаларында, Киршбраун теоремасы былай тұжырымдайды: егер U белгілі бір Гильберт кеңістігінің ішкі жиыны болса, ал V – басқа Гильберт кеңістігі болса, және
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space , and is another Hilbert space, and
is a Lipschitz continuous map, then there is a Lipschitz continuous map
that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean spaces and , and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient. The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of with the maximum norm and carries the Euclidean norm. More generally, the theorem fails for equipped with any norm (Schwartz 1969, p. 20). In general, an extension can also be written for valued functions as where and conv(g) is the lower convex envelope of g.
f : U → V – Липшиц үздіксіз бейнелеу болса, онда f-ті кеңейтетін және f-пен бірдей Липшиц тұрақтысына ие Липшиц үздіксіз бейнелеу
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space , and is another Hilbert space, and
is a Lipschitz continuous map, then there is a Lipschitz continuous map
that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean spaces and , and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient. The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of with the maximum norm and carries the Euclidean norm. More generally, the theorem fails for equipped with any norm (Schwartz 1969, p. 20). In general, an extension can also be written for valued functions as where and conv(g) is the lower convex envelope of g.
F : V → V бар.
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space , and is another Hilbert space, and
is a Lipschitz continuous map, then there is a Lipschitz continuous map
that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean spaces and , and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient. The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of with the maximum norm and carries the Euclidean norm. More generally, the theorem fails for equipped with any norm (Schwartz 1969, p. 20). In general, an extension can also be written for valued functions as where and conv(g) is the lower convex envelope of g.
Бұл нәтиже ерекше жағдайда Евклид кеңістіктеріне де қатысты, және Киршбраун осы формада теореманы тұжырымдап, дәлелдеген. Гильберт кеңістіктері үшін осы теореманың нұсқасын мысалы (Шварц, 1969, 21-беттен) табуға болады. Егер V – ажыратылатын кеңістік болса (әсіресе, егер ол Евклид кеңістігі болса), онда нәтиже Зермело-Франкель жиындар теориясында дұрыс; толық жалпы жағдай үшін таңдау аксиомасының кейбір түрі қажет болуы мүмкін; Бульдік жайқы идеал теоремасы жеткілікті екені белгілі. Теореманың дәлелі Гильберт кеңістіктерінің геометриялық ерекшеліктерін пайдаланады; Банах кеңістіктері үшін сәйкес мәлімдеме жалпы жағдайда, тіпті шекті өлшемді Банах кеңістіктері үшін де дұрыс емес. Мысалы, домен максималды нормаға ие және Евклид нормасын қамтитын U-дың ішкі жиыны болатын контр-мысалдарды құруға болады. Жалпы алғанда, теорема кез келген нормамен жабдықталған кеңістіктер үшін жарамсыз (Шварц, 1969, 20-бет). Жалпы, бағаланған функциялар үшін де кеңейтуді былай жазуға болады: , мұнда және conv(g) – g функциясының төменгі дөңес қабығы.
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space , and is another Hilbert space, and
is a Lipschitz continuous map, then there is a Lipschitz continuous map
that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean spaces and , and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient. The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of with the maximum norm and carries the Euclidean norm. More generally, the theorem fails for equipped with any norm (Schwartz 1969, p. 20). In general, an extension can also be written for valued functions as where and conv(g) is the lower convex envelope of g.
Тарих
Теорема Можез Дэвид Кирсбраун тарапынан дәлелденді, ал кейін Фредерик Валентин оны қайта дәлелдеді, ол алғаш рет Евклид жазықтығы үшін дәлелдеген болатын. Кейде бұл теорема Кирсбраун–Валентин теоремасы деп те аталады.