Asplund Space: Another Criterion

The theorem proved in this paper establishes conditions under which a Banach space $X$ is an Asplund space (i.e., its dual space is a space with the $RN$ property). The theorem is formulated in terms of the existence of a supersequentially compact set in $(B(X^{**}),\omega^*)$, where $B(X^{**})$ stands for the unit ball of the second dual of $X$ and $\omega^*$ for the weak topology on the ball. The example presented in the paper shows that one cannot get rid of some restrictive conditions in the theorem in general.