Approximation properties of the positive and negative parts of an element in ordered Banach spaces (Q1387405)

The real Banach space \(E\), ordered by a closed cone \(E_+\) is said to be regulary ordered if the following holds: (1) if \(x,y\in E\) and \(\pm x\leq y\), then \(\| x\|\leq\| y\|\), (2) for any \(x\in E\) there exists a \(y\in E_+\) such that \(\pm x\leq y\) and \(\| x\|= \| y\|\). The cone \(E_+\) is said to be attainable if for any \(x\in E\) there exists an element \(Px\in E_+\) on which the minimum in the formula for the distance from \(x\) to \(E_+\) is attained: \(d(x, E_+)= \inf\{\| a- x\|: a\in E_+\}\). The set of all elements \(Px\), for which \(d(x,E_+)= \| Px- x\|\) will be denoted by \(M(x)\). The following problem is studied in the paper: For which Banach spaces \(E\) does the relation \(M(x)= X_+\) hold for any \(x\in E\)? On the one hand, this property makes it possible to find the distance from the element to a cone and also the nearest points, on the other hand, this gives a metric characterization of the positive and negative part of an element. In the paper, the problem is completely solved for Hilbert spaces. For a more general class of spaces, the author suggests to define that an attainable cone \(E_+\) is said to be completely attainable if \(M(x)= X_+\) for any \(x\) and to be completely regular if for any \(x\in E\) and \(x_+, x_-\in E_+\), such that \(x= x_+- x_-\) and \(\| x_+- x_-\|=\| x_++ x_-\|\) the equality \(\| x_+- \lambda x_-\|= \| x_++ \lambda x_-\|\) holds for all real \(\lambda\). The author proves that any completely attainable cone is completely regular.