We show that a real normed linear space endowed with the $$\rho $$ -orthogonality relation, in general need not be an orthogonality space in the sense of Ratz. However, we prove that $$\rho $$ -orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.

中文翻译：

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在 $$\rho $$-正交可加映射上

我们证明了具有 $$\rho $$ -正交关系的实赋范线性空间，通常不需要是 Ratz 意义上的正交空间。然而，我们证明了在一些经典 Banach 空间上定义的 $$\rho $$ -正交可加映射必须是可加的。此外，考虑了近似正交条件下的可加性（和近似可加性）。