Ivan Shishkin, Rye (1878)

Problems/Linear algebraUnreviewed

Une caractérisation du projeté sur un convexe fermé

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This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
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These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Soit HH un espace de Hilbert et C⊂HC\subset H un sous-ensemble convexe fermé. Soient x,ξ∈Hx,\xi\in H. On veut montrer la caractérisation suivante de la projection sur CC, que l’on note PC:H→CP_C:H\to C :
ξ=PC(x)⟺∀y∈C,Re⁡(⟨ξ−x,ξ−y⟩)≤0.\xi=P_C(x) \quad\Longleftrightarrow\quad \forall y\in C,\qquad \operatorname{Re}\bigl(\langle \xi-x,\xi-y\rangle\bigr)\leq 0.

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