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 determining a Hilbert space. We use it on a finite dimensional real Banach space \({({\mathbb{X}, \| \cdot \|})}\) 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.
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The authors gratefully acknowledge the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
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Horváth, Á.G., Lángi, Z. & Spirova, M. Semi-Inner Products and the Concept of Semi-Polarity. Results Math 71, 127–144 (2017). https://doi.org/10.1007/s00025-015-0510-y
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DOI: https://doi.org/10.1007/s00025-015-0510-y