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The generalized decomposition theorem in Banach spaces and its applications. (English) Zbl 1067.46009

A Hilbert space can be decomposed as the product of a closed subspace \(K\) with its orthogonal complement. An analogous result is true if \(K\) is a closed convex cone and the role of the complement of \(K\) is held by the polar cone of \(K\). In [Appl. Math. Lett. 11, No. 6, 115–121 (1998; Zbl 0947.46012) and Field Inst. Commun. 25, 77–93 (2000; Zbl 0971.46004)], Y. Alber proved a form of a decomposition of a reflexive, strictly convex, smooth Banach space with respect to a closed convex cone. The authors present a simple proof of Alber’s decomposition, and then prove a new decomposition theorem for such Banach spaces. The decompositions are related to the best approximation operator, and the authors characterize when the generalized projection is the best approximation operator.

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces
41A50 Best approximation, Chebyshev systems
49J40 Variational inequalities
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