Approximative compactness in Musielak-Orlicz function spaces of Bochner type (Q728015)

Approximative compactness is an important property in the approximation theory. Recall that a nonempty subset \(C\) of a Banach space \(X\) is approximately compact if for each sequence \(\left( y_{n}\right) _{n=1}^{\infty }\) in \(C\) and any \(x\in X\) satisfying \(\left\| x-y_{n}\right\| \rightarrow \inf_{z\in C}\left\| x-z\right\| \) as \( n\rightarrow \infty ,\) the sequence \(\left( y_{n}\right) _{n=1}^{\infty }\) has a Cauchy subsequence. A Banach space \(X\) is said to be approximately compact whenever each nonempty, closed and convex subset of \(X\) is approximately compact. It is known that \(X\) is approximately compact if and only if it is reflexive and has the Kadec-Klee property. Moreover, the reflexivity is equivalent to the fact that each closed, convex subset \(C\) of \(X\) is proximinal. If the Banach space \(X\) is approximately compact and rotund, then the projector operator \(\pi _{C}\) is continuous. On the other hand, the geometry of Köthe-Bochner spaces has been developed intensively during the last decades. The authors study approximative compactness in the Musielak-Orlicz function spaces of Bochner type equipped with the Orlicz norm\ (denoted by \(L_{M}^{0}\left( X\right) \)). They find necessary and sufficient conditions for a proximinal convex subset of \(L_{M}^{0}\left( X\right) \) to be approximately compact (Theorem 2.1). As a consequence, they obtain a characterization of approximately compact Musielak-Orlicz function spaces of Bochner type (Corollary 2.9).