Approximative compactness and continuity of metric projector in Banach spaces and applications (Q931483)

A nonempty subset \(C\) of a normed space \(X\) is called approximatively compact if for every \(x\in X\) every minimizing sequence \(\{y_n\}\) in \(C\) (i.e., such that \(\| x-y_n\| \to d_C(x):=\inf \{\| x-y\| : y\in C\}\)) has a Cauchy subsequence. The space \(X\) is called midpoint locally uniformly rotund (MLUR) if, for every \(x,\{x_n\},\{y_n\}\) in \(S_X\) (the unit sphere of \(X\)), \(\lim_{n\to\infty}\| x_n+y_n-2x\| =0\) implies \(\lim_{n\to \infty}\| x_n-y_n\| =0\). The present paper is concerned with various relations between the geometric properties (smoothness and rotundity) of the normed space \(X\) and the approximation properties of its subsets. For instance, in an MLUR Banach space \(X\), a nonempty closed convex subset \(C\) of \(X\) is Chebyshev with continuous metric projection \(\pi_C\) if and only if \(C\) is approximatively compact (Theorem~15). The results are applied to study the Moore--Penrose metric generalized inverses of bounded linear operators \(T:X\to Y\) between Banach spaces \(X,Y\), as defined by \textit{H.\,Wang} and \textit{Y.--W.\thinspace Wang} [Chin.\ Ann.\ Math., Ser.\,B 24, No.\,4, 509--520 (2003; Zbl 1048.46022)].