Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces

TL;DR

This paper develops a duality theory for finite-dimensional near-vector spaces and introduces a generalized inner product, extending classical concepts to broader algebraic structures and complex datasets.

Contribution

It introduces a new inner product for strongly regular near-vector spaces and extends the theory of generalized means to complex data.

Findings

Established duality for finite-dimensional near-vector spaces

Defined a generalized inner product for strongly regular near-vector spaces

Unified classical means within a broader complex dataset framework

Abstract

In this paper we develop a duality theory for all finite-dimensional near-vector spaces and introduce a notion of inner product tailored to the broad and natural class of strongly regular near-vector spaces. This generalized construction extends the classical inner product beyond the classical framework, yielding rich families of examples on multiplicative near-vector spaces. Within this setting, several familiar norms-such as those that fail to produce Hilbert spaces in the classical sense-emerge naturally as genuine inner-product-type norms. A further contribution is the extension of the theory of generalized (weighted) means to arbitrary complex datasets. This extension unifies and generalizes the classical power and geometric means, carrying them beyond the domain of positive reals.