Convexities and approximative compactness and continuity of metric projection in Banach spaces

In this paper, we investigate the continuities of the metric projection in a nonreflexive Banach space X, which improve the results in X.N. Fang, J.H. Wang, Convexity and continuity of metric projection, Math. Appl. 14 (1) (2001) 47–51; P.D. Liu, Y.L. Hou, A convergence theorem of martingales in the limit, Northeast. Math. J. 6 (2) (1990) 227–234; H.J. Wang, Some results on the continuity of metric projections, Math. Appl. 8 (1) (1995) 80–84]. Under the assumption that X has some convexities, we discuss the relationship between approximative compactness of a subset A of X and continuity of the metric projection P_A. We also give a representation theorem for the metric projection to a hyperplane in dual space X^∗ and discuss its continuity.