Properties of the Space of Sections of Some Banach Bundles

TL;DR

This paper investigates the relationship between the topological properties of Banach bundle base spaces and the separability of their section spaces, establishing conditions under which these properties are equivalent or imply each other.

Contribution

It proves that for certain Banach bundles, base space second countability is equivalent to the separability of the section space, and characterizes Eberlein compactness via weakly compact generation.

Findings

Separable fibers and second countable base imply separable section space.

Separable section space implies second countability of the base space.

Section space generated by a weakly compact set implies the base is Eberlein compact.

Abstract

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 the all 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.