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
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| - Hilbert projection theorem (en)
- Teorema della proiezione (it)
- Théorème de projection sur un convexe fermé (fr)
- Twierdzenie o zbiorze wypukłym (pl)
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| - 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)
- In matematica, il teorema della proiezione o teorema della proiezione in spazi di Hilbert è 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 è vero per ogni sottospazio chiuso di : in tal caso una condizione necessaria e sufficiente per è che il vettore sia ortogonale a . (it)
- En mathématiques, le théorème de projection orthogonale sur un convexe fermé est un résultat de minimisation de la distance dont le principal corollaire est l'existence d'un supplémentaire orthogonal, donc d'une projection orthogonale sur un sous-espace vectoriel fermé. Dans le cadre particulier d'un espace de Hilbert, il remplace avantageusement le théorème de Hahn-Banach. Il est en effet plus simple à démontrer et plus puissant dans ses conséquences. (fr)
- Twierdzenie o zbiorze wypukłym – twierdzenie analizy funkcjonalnej mówiące, że każdy niepusty zbiór domknięty i wypukły w przestrzeni Hilberta zawiera jeden i tylko jeden element o najmniejszej normie. (pl)
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| - Proposition (en)
- Hilbert projection theorem (en)
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| proof
| - Let and be two minimum points. Then:
Since belongs to we have and therefore
Hence which proves uniqueness. (en)
- Assume that is a closed vector subspace of It must be shown the minimizer is the unique element in such that for every
Proof that the condition is sufficient:
Let be such that for all
If then and so
which implies that
Because was arbitrary, this proves that and so is a minimum point.
Proof that the condition is necessary:
Let be the minimum point. Let and
Because the minimality of guarantees that Thus
is always non-negative and must be a real number.
If then the map has a minimum at and moreover, which is a contradiction.
Thus (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:
and
Therefore
.
By 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:
The 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)
- Let be as described in this theorem and let
This theorem will follow from the following lemmas.
thumb|Vectors involved in the parallelogram law:
Because is convex, if then so that by definition of the infimum, which implies that
By the parallelogram law,
where now implies
and so
The 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
Since is complete, there exists some such that in
Because every belongs to which is a closed subset of their limit must also belongs to this closed subset, which proves that
Since the norm is a continuous function, in implies that in But also holds so that .
The existence of the sequence follows from the definition of the infimum, as is now shown.
The set is a non-empty subset of non-negative real numbers and
Let be an integer.
Because there exists some such that
Since holds . Thus and now the squeeze theorem implies that in .
For every the fact that means that there exists some such that
The convergence in thus becomes in
Lemma 2 and Lemma 1 together prove that there exists some such that
Lemma 1 can be used to prove uniqueness as follows.
Suppose 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
Because for every in which shows that the sequence satisfies the hypotheses of Lemma 1.
Lemma 1 guarantees the existence of some such that in
Because converges to so do all of its subsequences.
In particular, the subsequence converges to which implies that . Similarly, because the subsequence converges to both and Thus which proves the theorem. (en)
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| - Proof (en)
- Proof that is unique (en)
- Proof that a minimum point exists (en)
- Proof of characterization of minimum point when is a closed vector subspace (en)
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| has abstract
| - 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)
- En mathématiques, le théorème de projection orthogonale sur un convexe fermé est un résultat de minimisation de la distance dont le principal corollaire est l'existence d'un supplémentaire orthogonal, donc d'une projection orthogonale sur un sous-espace vectoriel fermé. Dans le cadre particulier d'un espace de Hilbert, il remplace avantageusement le théorème de Hahn-Banach. Il est en effet plus simple à démontrer et plus puissant dans ses conséquences. Il possède de nombreuses applications, en analyse fonctionnelle, en algèbre linéaire, en théorie des jeux, pour la modélisation mathématiques des sciences économiques ou encore pour l'optimisation linéaire. (fr)
- In matematica, il teorema della proiezione o teorema della proiezione in spazi di Hilbert è 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 è vero per ogni sottospazio chiuso di : in tal caso una condizione necessaria e sufficiente per è che il vettore sia ortogonale a . (it)
- Twierdzenie o zbiorze wypukłym – twierdzenie analizy funkcjonalnej mówiące, że każdy niepusty zbiór domknięty i wypukły w przestrzeni Hilberta zawiera jeden i tylko jeden element o najmniejszej normie. Wynik ten znajduje zastosowanie m.in. w dowodzie twierdzenia o rzucie ortogonalnym mającym swoje implikacje np. w rachunku prawdopodobieństwa (wykorzystywanym w jednym z dowodów istnienia warunkowej wartości oczekiwanej). Twierdzenie o zbiorze wypukłym lub, równoważnie, wynikające z niego daje stosunkowo krótki dowód twierdzenia Brouwera o punkcie stałym dla dowolnego przekształcenia zwartego zbioru wypukłego w siebie (-wymiarowa przestrzeń euklidesowa jest przestrzenią Hilberta). (pl)
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| math statement
| - If is a closed vector subspace of a Hilbert space then (en)
- For every vector in a Hilbert space and every nonempty closed convex there exists a unique vector for which is equal to
If the closed subset is also a vector subspace of then this minimizer is the unique element in such that is orthogonal to (en)
- For every nonempty closed convex subset of a Hilbert space there exists a unique vector such that
Furthermore, letting if is sequence in such that in then in (en)
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