A Cheney and Wulbert type lifting theorem in optimization

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Research Article

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Bollettino dell'Unione Matematica Italiana


In this paper we exhibit new classes of Banach spaces for which strong notions of optimization can be lifted from quotient spaces. Motivated by a well known result of Cheney and Wulbert on lifting of proximinality from a quotient space to a subspace, for closed subspaces, Z⊂ Y⊂ X, we consider stronger forms of optimization, that Z has in X and the quotient space Y / Z has in X / Z should lead to the conclusion Y has the same property in X. The versions we consider have been studied under various names in the literature as L-proximinal subspaces or subspaces that have the strong-112-ball property. We give an example where the strong-112-ball property fails to lift to the quotient. We show that if every M-ideal in Y is a M-summand, for a finite codimensional subspace Z⊂ Y, that is a M-ideal in X with the strong-112-ball property in X and if Y / Z has the 112-ball property in X / Z, then Y has the strong-112-ball property in X.

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