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Approximately bisectrix-orthogonality preserving mappings

100%

Zamani A.

Commentationes Mathematicae

2014

tom 54

nr 2

Regarding the geometry of a real normed space ${\mathcal X}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in {\mathcal X}$ as follows:$${x\sideset{ ^{\varepsilon\!\!}}{}\perp}_W y \mbox{~if and only if~} \sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq \Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$ We study the class of linear mappings preserving the approximately bisectrix-orthogonality ${\sideset{ ^{\varepsilon\!\!}}{}\perp}_W$. In particular, we show that if $T: {\mathcal X}\to {\mathcal Y}$ is an approximate linear similarity, then $${x\sideset{ ^{\delta\!\!}}{}\perp}_W y\Longrightarrow {Tx \sideset{ ^{\theta\!\!}}{}\perp}_W Ty \qquad (x, y\in {\mathcal X})$$ for any $\delta\in[0, 1)$ and certain $\theta\geq 0$.

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1 Commentationes Mathematicae

1 Zamani A.

1 2014

Wyniki wyszukiwania

Approximately bisectrix-orthogonality preserving mappings