Finitely locally finite coverings of Banach spaces

A well known result due to H. Corson
states that, for any covering $\tau$ by closed bounded convex
subsets of any Banach space $X$ containing an infinite-dimensional
reflexive subspace, there exists a compact subset $C$ of $X$ that
meets infinitely many members of $\tau$. We strengthen this result
proving that, even under the weaker assumption that $X$ contains an
infinite-dimensional separable dual space, a (algebraically)
finite-dimensional compact set $C$ with that property can always be
found.