Banach spaces with the (strong) Gelfand--Phillips property

Abstract

Several new characterizations of the Gelfand-Phillips property are given. We define a strong version of the Gelfand-Phillips property and prove that a Banach space has this stronger property iff it embeds into $c_0$. For an infinite compact space $K$, the Banach space $C(K)$ has the strong Gelfand-Phillips property iff $C(K)$ is isomorphic to $c_0$ iff $K$ is countable and has finite scattered height.

10 pages. arXiv admin note: substantial text overlap with arXiv:2003.06764