such that $\|Tx_n\| \longrightarrow \|T\|$ and $\langle Tx_n, Ax_n \rangle \longrightarrow 0$. If the space is finite dimensional it then follows that $T \bot_B A$ if and only if there exists $ x \in S_\mathbb{H}$ such that $ \|Tx\| = \|T\| $ and $  \langle Tx, Ax \rangle = 0.$ Recently in [@S], Sain characterized the Birkhoff-James orthogonality of linear operators on a finite dimensional real Banach space . The following two definitions were necessary to obtain the desired characterization in [@S]:\

Let $T \in \mathbb{B}(\mathbb{X}, \mathbb{Y})$. We define $M_T$ to be the set of all unit vectors in $S_\mathbb{X}$ at which $T$ attains norm, i.e., $$M_T = \{ x \in S_\mathbb{X} \colon \|Tx\| = \|T\| \}.$$

For any two elements $ x, y $ in a real normed linear space $ \mathbb{X}, $ let us say that $ y \in x^{+} $ if $ \| x + \lambda y \| \geq \| x \| $ for all $ \lambda \geq 0. $ Accordingly, we say that $ y \in x^{-} $ if $ \| x + \lambda y \| \geq \| x \| $ for all $ \lambda \leq 0$.

In this paper, our aim is to extend the 