Universal measurability of the identity mapping of a Banach space in certain topologies

If X is a Banach space and X is its conjugate, then a subset Y of X is called madmissible for X if a) the topology (X, Y) is Hausdorff, b) the identity embedding of (X, (X, Y)) into X is universally measurable (Ref. Zh. Mat., 1975, 8B 75 8K). If X is separable, then the existence of an m-admissible set is well known. In this note it is shown that there exist nonseparable X having separable m-admissible sets. The properties of spaces with separable m-admissible sets are considered. It is proved, in particular, that a separable normalizing subset Y of X is m-admissible for X if and only if every (X, Y)-compact set is separable in X.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 305–314, February, 1978.