Semi-Inner Products and the Concept of Semi-Polarity

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• 作者：Ákos G. Horváth ; Zsolt Lángi ; Margarita Spirova
• 关键词：Antinorm ; gauge function ; isoperimetrix ; Minkowski space ; normality ; normed space ; polarity ; semi ; inner product ; symplectic bilinear form ; support function
• 刊名：Results in Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：71
• 期：1-2
• 页码：127-144
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-9012
• 卷排序：71

文摘

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.