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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)], {\it 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:
46B10Duality and reflexivity in normed spaces
41A50Best approximation, Chebyshev systems
49J40Variational methods including variational inequalities
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Full Text: DOI
References:
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