Quasi-Chebyshev subspaces in dual spaces (Q2739107)

If each \(x\) in a Banach space \(X\) has a unique best approximation in a linear subspace \(W\), then \(W\) is a Chebyshev subspace of \(X\). \(W\) is called quasi-Chebyshev if \(P_w(x)= \{y\in W:\|x-y\|= d(x,W)\}\) is a non-empty and compact set in \(X\) for every \(x\in X\).NEWLINENEWLINENEWLINEVarious characterizations of weak\(^*\)-closed quasi-Chebyshev subspaces in dual Banach spaces are proven. Also, a characterization is given of the dual space of a Banach space in which all weak\(^*\)-closed linear subspaces are quasi-Chebyshev.