"Assume that is a closed vector subspace of It must be shown the minimizer is the unique element in such that for every \n \nProof that the condition is sufficient: \nLet be such that for all \nIf then and so \n \nwhich implies that \nBecause was arbitrary, this proves that and so is a minimum point. \n\nProof that the condition is necessary: \nLet be the minimum point. Let and \nBecause the minimality of guarantees that Thus\n\nis always non-negative and must be a real number. \nIf then the map has a minimum at and moreover, which is a contradiction. \nThus"@en . "For every nonempty closed convex subset of a Hilbert space there exists a unique vector such that \n\nFurthermore, letting if is sequence in such that in then in"@en . . . . . . "Twierdzenie o zbiorze wypuk\u0142ym \u2013 twierdzenie analizy funkcjonalnej m\u00F3wi\u0105ce, \u017Ce ka\u017Cdy niepusty zbi\u00F3r domkni\u0119ty i wypuk\u0142y w przestrzeni Hilberta zawiera jeden i tylko jeden element o najmniejszej normie. Wynik ten znajduje zastosowanie m.in. w dowodzie twierdzenia o rzucie ortogonalnym maj\u0105cym swoje implikacje np. w rachunku prawdopodobie\u0144stwa (wykorzystywanym w jednym z dowod\u00F3w istnienia warunkowej warto\u015Bci oczekiwanej). Twierdzenie o zbiorze wypuk\u0142ym lub, r\u00F3wnowa\u017Cnie, wynikaj\u0105ce z niego daje stosunkowo kr\u00F3tki dow\u00F3d twierdzenia Brouwera o punkcie sta\u0142ym dla dowolnego przekszta\u0142cenia zwartego zbioru wypuk\u0142ego w siebie (-wymiarowa przestrze\u0144 euklidesowa jest przestrzeni\u0105 Hilberta)."@pl . . "hidden"@en . . . . "Let be as described in this theorem and let\n\nThis theorem will follow from the following lemmas.\n\n\n\n\nthumb|Vectors involved in the parallelogram law: \nBecause is convex, if then so that by definition of the infimum, which implies that \nBy the parallelogram law,\n\nwhere now implies \n\nand so\n\nThe assumption implies that the right hand side of the above inequality can be made arbitrary close to by making and sufficiently large. The same must consequently also be true of the inequality's left hand side and thus also of which proves that is a Cauchy sequence in \n\nSince is complete, there exists some such that in \nBecause every belongs to which is a closed subset of their limit must also belongs to this closed subset, which proves that \nSince the norm is a continuous function, in implies that in But also holds so that . \n\n\n\n\n\n\n\nThe existence of the sequence follows from the definition of the infimum, as is now shown. \nThe set is a non-empty subset of non-negative real numbers and \nLet be an integer. \nBecause there exists some such that \nSince holds . Thus and now the squeeze theorem implies that in . \n\nFor every the fact that means that there exists some such that \nThe convergence in thus becomes in \n\n\n\nLemma 2 and Lemma 1 together prove that there exists some such that \nLemma 1 can be used to prove uniqueness as follows. \nSuppose is such that and denote the sequence by so that the subsequence of even indices is the constant sequence while the subsequence of odd indices is the constant sequence \nBecause for every in which shows that the sequence satisfies the hypotheses of Lemma 1. \nLemma 1 guarantees the existence of some such that in \nBecause converges to so do all of its subsequences. \nIn particular, the subsequence converges to which implies that . Similarly, because the subsequence converges to both and Thus which proves the theorem."@en . . . . "Twierdzenie o zbiorze wypuk\u0142ym \u2013 twierdzenie analizy funkcjonalnej m\u00F3wi\u0105ce, \u017Ce ka\u017Cdy niepusty zbi\u00F3r domkni\u0119ty i wypuk\u0142y w przestrzeni Hilberta zawiera jeden i tylko jeden element o najmniejszej normie."@pl . . . . . . "For every vector in a Hilbert space and every nonempty closed convex there exists a unique vector for which is equal to \n\nIf the closed subset is also a vector subspace of then this minimizer is the unique element in such that is orthogonal to"@en . . . . . "Let be the distance between and a sequence in such that the distance squared between and is less than or equal to Let and be two integers, then the following equalities are true:\n\nand\n\nTherefore\n\n.\n\nBy giving an upper bound to the first two terms of the equality and by noticing that the middle of and belong to and has therefore a distance greater than or equal to from it follows that:\n\n\nThe last inequality proves that is a Cauchy sequence. Since is complete, the sequence is therefore convergent to a point whose distance from is minimal."@en . . . . . "Proof of characterization of minimum point when is a closed vector subspace"@en . . . . . . "If is a closed vector subspace of a Hilbert space then"@en . "Proof that a minimum point exists"@en . . . "Proof"@en . . . "true"@en . "In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector in a Hilbert space and every nonempty closed convex there exists a unique vector for which is minimized over the vectors ; that is, such that for every"@en . . "Proposition"@en . . . . . "Teorema della proiezione"@it . "Let and be two minimum points. Then:\n\n\nSince belongs to we have and therefore\n\n\nHence which proves uniqueness."@en . . "Hilbert projection theorem"@en . "1070622844"^^ . "Twierdzenie o zbiorze wypuk\u0142ym"@pl . . "In matematica, il teorema della proiezione o teorema della proiezione in spazi di Hilbert \u00E8 un risultato dell', utilizzato spesso in analisi funzionale, che stabilisce che per ogni punto in uno spazio di Hilbert e per ogni insieme convesso chiuso esiste un unico tale per cui la distanza assume il valore minimo su . In particolare, questo \u00E8 vero per ogni sottospazio chiuso di : in tal caso una condizione necessaria e sufficiente per \u00E8 che il vettore sia ortogonale a ."@it . . . . "In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector in a Hilbert space and every nonempty closed convex there exists a unique vector for which is minimized over the vectors ; that is, such that for every"@en . . . . "9644792"^^ . . "En math\u00E9matiques, le th\u00E9or\u00E8me de projection orthogonale sur un convexe ferm\u00E9 est un r\u00E9sultat de minimisation de la distance dont le principal corollaire est l'existence d'un suppl\u00E9mentaire orthogonal, donc d'une projection orthogonale sur un sous-espace vectoriel ferm\u00E9. Dans le cadre particulier d'un espace de Hilbert, il remplace avantageusement le th\u00E9or\u00E8me de Hahn-Banach. Il est en effet plus simple \u00E0 d\u00E9montrer et plus puissant dans ses cons\u00E9quences. Il poss\u00E8de de nombreuses applications, en analyse fonctionnelle, en alg\u00E8bre lin\u00E9aire, en th\u00E9orie des jeux, pour la mod\u00E9lisation math\u00E9matiques des sciences \u00E9conomiques ou encore pour l'optimisation lin\u00E9aire."@fr . . "Proof that is unique"@en . . . . "22473"^^ . "En math\u00E9matiques, le th\u00E9or\u00E8me de projection orthogonale sur un convexe ferm\u00E9 est un r\u00E9sultat de minimisation de la distance dont le principal corollaire est l'existence d'un suppl\u00E9mentaire orthogonal, donc d'une projection orthogonale sur un sous-espace vectoriel ferm\u00E9. Dans le cadre particulier d'un espace de Hilbert, il remplace avantageusement le th\u00E9or\u00E8me de Hahn-Banach. Il est en effet plus simple \u00E0 d\u00E9montrer et plus puissant dans ses cons\u00E9quences."@fr . "Th\u00E9or\u00E8me de projection sur un convexe ferm\u00E9"@fr . . . . . . . . "Hilbert projection theorem"@en . "In matematica, il teorema della proiezione o teorema della proiezione in spazi di Hilbert \u00E8 un risultato dell', utilizzato spesso in analisi funzionale, che stabilisce che per ogni punto in uno spazio di Hilbert e per ogni insieme convesso chiuso esiste un unico tale per cui la distanza assume il valore minimo su . In particolare, questo \u00E8 vero per ogni sottospazio chiuso di : in tal caso una condizione necessaria e sufficiente per \u00E8 che il vettore sia ortogonale a ."@it . .