Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way. It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X, then the metric projector pi(C) from X onto C is continuous. Under the assumption that X is midpoint locally uniformly rotund, we prove that the approximative compactness of C is also necessary for the continuity of the projector pi(C) by the method of geometry of Banach spaces. Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T+, where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.