Almost Free Non-Archimedean Banach Spaces and Relation to Large Cardinals

TL;DR

This paper explores the structure of non-Archimedean Banach spaces, showing that almost free Banach spaces are free under certain large cardinal assumptions, paralleling classical Abelian group results.

Contribution

It introduces the concept of almost free Banach $k$-vector spaces and proves their freeness under large cardinal assumptions, extending classical Abelian group theory.

Findings

Almost free Banach $k$-vector spaces are free under $eth_1$-strong compactness.

The work establishes non-Archimedean analogues of classical group-theoretic facts.

The results connect large cardinal axioms with the structure of Banach spaces.

Abstract

Let $k$ be a complete valuation field. We formulate a free Banach $k$-vector space as a Banach $k$-vector space with an orthonormal Schauder basis, and an almost free Banach $k$-vector space as a non-Archimedean analogue of an almost free Abelian group. As non-Archimedean analogues of the classical facts that an almost free Abelian group is free under the assumption of the $\aleph_1$-strong compactness or the weak compactness of the cardinality, we show that an almost free Banach $k$-vector space is free under similar assumptions.