Continuity of metric projections onto subspaces and openness of quotient maps on unit balls (Q1069083)

Let M be a proximinal linear subspace of a normed linear space X (i.e., for any \(x\in X\) there is a best approximation of x in M). The corresponding best approximation operator \(P: X\to M\) is multivalued in general. From the main result of the paper it follows: Let P be single-valued; then P is continuous if and only if the quotient mapping \(Q: X\to X/M\) is relatively open on the closed unit ball of X. A more general result concerning multivalued operator P is proved (then, the continuity of P is to be replaced by the lower semicontinuity). The main theorem is applied to obtain some results in both sequence and function spaces. Also, the pointwise Lipschitz continuity of P is discussed. The applications presented in the paper show the merit of the main theorem especially in the case that M is of a finite codimension in X since then the quotient map Q is easy to represent.