Some results on Singer orthogonality and characterizations of inner product spaces
Let \(E\) be a real normed linear space. Two vectors \(x,y\in E\) are said to be Singer orthogonal [\textit{I. Singer}, Acad. R. P. Romine. Bul. Sţi. Secţ. Mat. Fiz. 9, 29-42 (1957; Zbl 0088.318)], \((x\perp{}{_ s y})\), if either \(\| x\| \| y\|=0\) or \(\| x/\| x\|+ y/\| y\| \|= \| x/\| x\|-y/ \| y\| \|\). Let \(S\) be the unit sphere of \(E\). The following properties are proved: 1) If \(x,y\in S\), \(\alpha\in \mathbb{R}\) and \(x\perp{}{_ s \alpha}x+y\) then \(|\alpha|\leq (1+\sqrt {2})/2\); 2) For every \(x,y\in E\) there exists a unique \(\beta\in\mathbb{R}\) such that \(x+\beta y\perp{}{_ s x}-\beta y\) (moreover, \(1/\sqrt{2} \leq |\beta| \leq \sqrt{2}\) if \(x,y\in S\)). It is well known that if the norm is induced by an inner product then Singer orthogonality coincides with usual orthogonality and \(|\alpha|\) in property 1) is bounded by 1. Conversely, it is proved that this bound is characteristic of inner product spaces. On the other hand, if \(E\) is an inner product space and \(x,y\in S\) then \(x+y\perp{}{_ sx}-y\). This property is also characteristic of such spaces.
scientific article; zbMATH DE number 446284