Gelfand--Phillips type properties of locally convex spaces

Abstract

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of the Gelfand--Phillips property and the (coarse) Gelfand--Phillips property of order $p$ of Banach spaces, we introduce and study different types of the Gelfand--Phillips property of order $(p,q)$ (the $GP_{(p,q)}$ property) and the coarse Gelfand--Phillips property of order $p$ in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the $GP_{(p,q)}$ property. Characterizations of locally convex spaces with the introduced Gelfand--Phillips type properties are given.