Abstract
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, \(Z \subset Y \subset X\), 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-\(1\frac{1}{2}\)-ball property. We give an example where the strong-\(1\frac{1}{2}\)-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 \(Z \subset Y\), that is a M-ideal in X with the strong-\(1\frac{1}{2}\)-ball property in X and if Y / Z has the \(1\frac{1}{2}\)-ball property in X / Z, then Y has the strong-\(1\frac{1}{2}\)-ball property in X.
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Rao, T.S.S.R.K. A Cheney and Wulbert type lifting theorem in optimization. Boll Unione Mat Ital 10, 585–589 (2017). https://doi.org/10.1007/s40574-016-0090-0
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DOI: https://doi.org/10.1007/s40574-016-0090-0