Supplement to the paper “On the duality of functors generated by spaces of vector-valued functions”

Let $E$ be a Banach space of measurable functions, and $X$ a Banach space. It is known that $E\otimes X$ is dense in the space $E(X)$ of vector-valued functions if the condition (A) holds: $(e_n\downarrow0)\Rightarrow(\|e_n\|\to0)$. The necessity of this condition was shown in the paper cited in the title (RZhMat., 1976, 5B740) under the assumption that $X$ contains a complemented infinite-dimensional subspace with unconditional basis. In the present paper the requirement of the existence of such a subspace is removed. Also, an error in the earlier paper is corrected.
Bibliography: 7 titles.