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 projectorπC 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 projectorπC 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.

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 π _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 π _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. This work was supported by the State Committee for Scientific Research, Poland (Grant No. 1P03A1127) and the National Nature Science Foundation of China (Grant Nos. 10471032, 10671049)