PROPERTIES OF THE SPACE OF SECTIONS OF SOME BANACH BUNDLES

One shows for Banach bundles in a certain class that having a second countable locally compact Hausdorff base space and separable fibers implies the separability of the Banach space of all the sections that vanish at infinity. In the reverse direction, it is proved that for a Banach bundle with locally compact Hausdorff base space the separability of the space of all the sections that vanish at infinity implies that the base space is second countable. If the base space is compact and the space of all the sections of the bundle is generated by a weakly compact subset then the base space is an Eberlein compact.