Semi-inner products and the concept of semi-polarity

TL;DR

This paper introduces semi-inner products in Banach spaces to generalize concepts like antinorms and polarity, enabling new geometric analyses in spaces lacking inner product symmetry.

Contribution

It develops a framework for semi-inner products in finite-dimensional Banach spaces, extending antinorms and defining semi-polarity concepts.

Findings

Generalization of antinorms to even-dimensional spaces

Introduction of normality maps in Banach spaces

Development of semi-polarity as a variant of polarity

Abstract

The lack of an inner product structure in Banach spaces yields the motivation to introduce a semi-inner product with a more general axiom system, one missing the requirement for symmetry, unlike the one determing a Hilbert space. We use it on a finite dimensional real Banach space $(\X, \| \cdot\|)$ to define and investigate three concepts. First, we generalize that of \emph{antinorms}, already defined in Minkowski planes, for even dimensional spaces. Second, we introduce \emph{normality maps}, which in turn leads us to the study of \emph{semi-polarity}, a variant of the notion of polarity, which makes use of the underlying semi-inner product.