On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces

TL;DR

This paper investigates the structure of quasi-normed spaces, demonstrating the existence of weaker topologies under certain conditions and providing new insights into basic sequences and dual space properties in quasi-Banach spaces.

Contribution

It establishes a new criterion for the existence of basic sequences in quasi-Banach spaces based on the presence of a subspace with a separating dual, and constructs explicit weaker topologies.

Findings

Quasi-normed spaces with a countable dimensional subspace with a separating dual admit a weaker Hausdorff topology.

A quasi-Banach space contains a basic sequence iff it has an infinite countable dimensional subspace with a separating dual.

All countable-dimensional subspaces of Kalton's minimal quasi-Banach space lack a separating family of continuous linear functionals.

Abstract

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by Kalton. Namely, which all of its $\aleph_0$-dimensional subspaces fail to have a separating family of continuous linear functionals.