Gelfand--Phillips type properties of locally convex spaces

TL;DR

This paper introduces and analyzes new Gelfand--Phillips type properties for locally convex spaces, extending classical Banach space notions and exploring their stability and characterizations.

Contribution

It defines Gelfand--Phillips type properties of order (p,q) for locally convex spaces and studies their stability, quotient relations, and characterizations.

Findings

Properties are stable under direct products, sums, and closed subspaces.

Any locally convex space is a quotient of one with the GP_{(p,q)} property.

Characterizations of spaces with these properties are provided.

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.