Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces
Leeandro Boonzaaier, Sophie Marques, Daniella Moore
TL;DR
This work generalizes core linear-algebraic concepts to strongly regular near-vector spaces by developing a flexible inner product framework and a duality theory that extend dual spaces, orthogonality, and angles beyond classical vector spaces. Through a carefully constructed set of deformed addition rules and multiplicative automorphisms, the authors recover classical norms such as $oldsymbol{ ext{ℓ}}^p$, $oldsymbol{ ext{ℓ}}^{p,q}$, and $oldsymbol{ ext{L}}^p$ as inner-product norms, and they further extend these ideas to infinite-norm regimes via limiting constructions in non-associative target fields. The paper then leverages this framework to define weighted generalized means for complex data, unifying power means, geometric means, and their complex-parameter counterparts, with potential applications in data analysis and beyond. Overall, the approach provides a versatile algebraic-analytic toolkit for analyzing near-vector spaces and extending geometric notions like angle, orthogonality, and duality to non-classical settings, while offering new means and norms applicable to complex and non-linear data structures.
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.