numbers $x$, $y$ and for any $\epsilon \in [0,1) $, $y\in x^{+(\epsilon)}$ if and only if $x y\geq 0$. Similarly, $y\in x^{- (\epsilon)}$ if and only if $x y\leq 0$.

Now, the promised characterization:

\[theorem:functional\] Let $\mathbb{X}$ be a normed linear space such that $\mathbb{X}^*$ is strictly convex. Then for any $f, g\in \mathbb{X}^*$, $f\bot_{B} g $ if and only if there exist $\{x_n\}$, $\{y_n\} $ in $ S_\mathbb{X}$ such that\
$(i)|f(x_n)|\rightarrow \|f\|$ and $ |f(y_n)| \rightarrow \|f\|$ as $n \rightarrow \infty$\
$(ii)f(x_n).g(x_n) \geq 0$ and $f(y_n).g(y_n) \leq 0$ for all $n\in \mathbb{N}$.

Since $ \mathbb{X}^{*} $ is strictly convex, $ f, g $ are rotund points of $ \mathbb{X}^{*}. $ The rest of the proof follows directly from Theorem  \[theorem:rotund\] and Proposition  \[prop:real\].

Similarly as an application of Theorem \[theorem:bounded\], Birkhoff-James orthogonality of bounded linear functionals on an infinite dimensional normed linear space can be stated as follows.

\[theorem:bounded functional\] Let $\mathbb{X}$ be a normed linear space. Then for any $f, g\in \mathbb{X}^*$, $f\bot_{B} g $ if and only if either of the conditions in (a) or in (b) holds:\
(a) there exists $\{x_n\}$ in $ S_\mathbb{X}$ such that $\mid f(x_n)\mid \longrightarrow \|f\| $ and $g(x_n) \longrightarrow 0.$\
(b) 