Norm Hilbert spaces over G-modules with a convex base

TL;DR

This paper introduces a new class of Norm Hilbert spaces over G-modules with convex bases, expanding the understanding of their structure and properties beyond classical and rank one cases.

Contribution

It defines and analyzes Norm Hilbert spaces where the G-module of norms has a convex base, a novel class not previously studied.

Findings

Characterization of spaces with convex base G-modules

Extension of classical Hilbert space concepts to G-modules

Insights into the structure of norms in these spaces

Abstract

By analogy with the classical definition, a Norm Hilbert space $E$ is defined as a Banach space over a valued field $K$ in which each closed subspace has an orthocomplement. In the rank one case (that is, the value group as well as the set of norms of the space are contained in $[0, \infty)$), they were described by van Rooij in his classical book of 1978, but the name itself was introduced in 1999 by Ochsenius and Schikhof for the case of spaces with an infinite rank valuation. Here we shall also consider only value groups that are contained in $(\R^+,\cdot)$, yet we borrow from the infinite rank case the notion of a $G$-module for the set of norms of the space. Their structure allows for greater complexity than that of ordered subsets of $\R$. In this paper we describe a new class of Norm Hilbert spaces, those in which the $G$-module has a convex base. Their characteristics will be the focus of our study.