We study symmetric points with respect to $(\rho_+)$-orthogonality, $(\rho_{ -})$-orthogonality and $\rho$-orthogonality in the space $C(K, \mathbb{ X}),$ where $K$ is a perfectly normal, compact space and $ \mathbb X$ is a Banach space. We characterize left symmetric points and right symmetric points in $C(K, \mathbb{ X})$ with respect to $(\rho_{ +})$-orthogonality and $(\rho_{ -})$-orthogonality, separately. Furthermore, we provide necessary conditions for left symmetric and right symmetric points with respect to $\rho$-orthogonality. As an application of these results we also study these symmetric points in the space of operators defined on some special Banach spaces.
Categories: math.FA