On (V) and (V*) sets in vector valued function spaces

Abstract

A. Pełczyński [Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641–648; defined and studied the properties (V) and (V*), proving that if a Banach space E [resp. its conjugate E∗] has property (V) then E∗ [resp. E] has property (V*), and asking whether the converse implications are true. It is known that the answer to Pełczyński's question is negative; E. Saab and P. Saab, Pacific J. Math. 125 (1986), no. 1, 205–210. The first part of this paper is concerned with the problem of characterizing Banach spaces for which the answer to Pełczyński's question is positive. The second part is devoted to a study of (V)- and (V*)-subsets and the heredity of (V)- and (V*)-properties for the Banach space C(Ω,E).