On the finite topology of a vector space and the domination problem for families of norms

TL;DR

This paper explores the structure of the finite topology on vector spaces, showing it equals the union of all normed topologies in countably dimensional spaces, and characterizes spaces where all norm families are dominated.

Contribution

It provides a characterization of vector spaces where every family of norms with a given cardinality is dominated, and clarifies the relationship between finite topology and normed space topologies.

Findings

Finite topology equals the union of all normed space topologies in countably dimensional spaces.

Characterization of vector spaces with dominated families of norms.

Finite topology coincides with the greatest vector space and locally convex topologies in countable dimensions.

Abstract

Let $V$ be a real or complex vector space. The finite topology of $V$ consists of all the subsets $U$ for which the intersection $U \cap F$ is closed in $F$ for every finite-dimensional linear subspace of $V$. It is known that if $V$ has countable dimension, then this topology coincides with the greatest vector space topology and with the greatest locally convex vector space topology. Here, we note that in this case the finite topology is simply the union of all the normed space topologies. In connection to this problem, we characterize the vector spaces on which every family of norms with given cardinality is dominated.