Let X be a Banach space such that there exists a Banach space \(^*X\) satisfying \(( ^*X )^ *= X\) . In this paper, we introduce X-valued Bourgain–Morrey spaces. We show that \(^*X\) -valued block spaces are the predual of X-valued Bourgain–Morrey spaces. We obtain the completeness, denseness, and Fatou property of \(^*X\) -valued block spaces. We give a description of the dual of X-valued Bourgain–Morrey spaces and conclude the reflexivity of these spaces. The boundedness of powered Hardy–Littlewood maximal operator in vector-valued block spaces is obtained.
