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.


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


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