it is well known that if $ \mathbb{X} $ is not reflexive then there exists a bounded linear functional on $ \mathbb{X} $ such that the functional does not attain norm. It is particularly in these cases that Theorem  \[theorem:functional\] can be effectively applied, provided the dual space $ \mathbb{X}^{*} $ is strictly convex. We would also like to note that there exists a non-reflexive Banach space whose dual is strictly convex. Indeed, the classical $ c_0 $ space (with the *sup* norm) has an unconditional basis and it does not contain a copy of $ l_1. $ Therefore, as pointed out in [@ST], there exists an equivalent renorming of this space with a strictly convex dual norm. Furthermore, we note that if a Banach space is reflexive under a particular norm, then it is reflexive under any equivalent norm. Combining these observations, we may and do conclude that there exists a norm on $ c_0 $ such that the space is nonreflexive and the dual space is strictly convex.

The next two theorems show how Birkhoff-James orthogonality($ T \bot_B A$) and strong Birkhoff-James orthogonality($T \bot_{SB}A$) are related in the space of bounded linear operators. A scalar $\lambda $ is 