• Journal of applied mathematics & informatics
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• v.42 no.5
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• pp.1155-1170
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• 2024

The concept of orthogonality is widely used in various fields of study, both within and outside the scope of mathematics, especially algebra. The concept of orthogonality gives a picture of the relationship between two vectors that are perpendicular to each other, or the inner product in both of them is zero. However, the concept of orthogonality has undergone significant development. One of the developments is Pythagorean orthogonality. In this paper, it is explored topics related to Pythagorean orthogonality and linear mappings in inner product spaces. It is also examined how linear mappings preserve Pythagorean orthogonality and provides insights into how mathematical transformations affect geometric relationships. The results reveal several properties that apply to linear mappings preserving Pythagorean orthogonality.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.157-167
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• 2008

A mapping f : $M{\rightarrow}N$ between Hilbert $C^*$-modules approximately preserves the inner product if $$\parallel<f(x),\;f(y)>-<x,y>\parallel\leq\varphi(x,y)$$ for an appropriate control function $\varphi(x,y)$ and all x, y $\in$ M. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers-Ulam-Rassias stability of the orthogonality equation.