Norm derivatives and geometry of bilinear operators

We study the norm derivatives in context of Birkhoff–James orthogonality real Banach spaces. As an application this, we obtain a complete characterization left-symmetric points and right-symmetric space terms derivatives. strong $$\ell _1^n$$ _\infty ^n$$ also relation defined by some newly introduced variation orthogonality. further orthogonality, approximate smoothness attainment bounded bilinear operators between