The generalized sine function and geometrical properties of normed spaces

Opuscula Mathematica

Tomasz Szostok¹ 

[1] Silesian University, Institute of Mathematics, Bankowa 14, 40-007 Katowice, Poland;

关键词: geometry of normed spaces;    smoothness;    strict convexity;    Birkhoff-James orthogonality;    conditional functional equations;   

DOI  :  http://dx.doi.org/10.7494/OpMath.2015.35.1.117

来源: DOAJ

【 摘 要 】

Let \((X,\|\cdot\|)\) be a normed space. We deal here with a function \(s:X\times X\to\mathbb{R}\) given by the formula \[s(x,y):=\inf_{\lambda\in\mathbb{R}}\frac{\|x+\lambda y\|}{\|x\|}\] (for \(x=0\) we must define it separately). Then we take two unit vectors \(x\) and \(y\) such that \(y\) is orthogonal to \(x\) in the Birkhoff-James sense. Using these vectors we construct new functions \(\phi_{x,y}\) which are defined on \(\mathbb{R}\). If \(X\) is an inner product space, then \(\phi_{x,y}=\sin\) and, therefore, one may call this function a generalization of the sine function. We show that the properties of this function are connected with geometrical properties of the normed space \(X\).

【 授权许可】

Unknown