Semi-Inner Products and the Concept of Semi-Polarity

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 determining a Hilbert space. We use it on a finite dimensional real Banach space ( X , ‖ · ‖ ) to define and investigate three concepts. First, we generalize that of antinorms , already defined in Minkowski planes, for even dimensional spaces. Second, we introduce normality maps , which in turn leads us to the study of semi-polarity , a variant of the notion of polarity, which makes use of the underlying semi-inner product.