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.