& \leq & \|Tx\| + \frac{1}{2} + \sum_{n=2}^{\infty}(\frac{1}{n^4}-\frac{1}{n^2})\\
                        & < & \|Tx\| - 0.06\\
                        & \leq & \|T\|-0.06.                        \end{aligned}$$ This shows that there does not exist $\{\epsilon_n\} $ with $\epsilon_n \longrightarrow 0 $ and $\{x_n\} \subset S_{\ell_1}$ such that $\|Tx_n\| \longrightarrow \|T\| $ and $Ax_n \in (Tx_n)^{+(\epsilon_n)}, $ for otherwise $$\sqrt{1 -{\epsilon_n }^2}||Tx_n\| \leq \|Tx_n + Ax_n\| < \|Tx_n\|-0.06,$$ and so letting $n \longrightarrow \infty$ we get a contradiction. Thus $T \bot_B A$ but $T$ does not satisfy hypothesis of Theorem \[theorem:rotund\].

The last example indicates that some additional condition is required to obtain the most generalized characterization of Birkhoff-James orthogonality of bounded linear operators defined between infinite dimensional normed linear spaces. We accomplish the goal in the next theorem.

\[theorem:bounded\] Let $\mathbb{X}$ and $\mathbb{Y}$ be two normed linear spaces. Let $T \in \mathbb{B} (\mathbb{X}, \mathbb{Y})$ be nonzero. Then for any $A \in \mathbb{B} (\mathbb{X}, \mathbb{Y}) ,~ T \bot_B A$ if and only if either of the conditions in $(a)$ or in $(b)$ holds:\
$(a)$ There exists a sequence $\{x_n\}$ in $S_\mathbb{X}$ such that $\| Tx_n\| \rightarrow \|T\|$ and $\|Ax_n\| \rightarrow 0$ as $n\rightarrow \infty$.\
$(b)$ There exists two sequences $\{x_n\},~\{y_n\}$ in $S_\mathbb{X}$ and two sequences of positive real numbers 