Approximate B-J Orthogonality

In order to describe the analytic structure and the geometry of an inner product space \((\mathbb {H}, \langle , \rangle ),\) the usual orthogonality relation \(``x\perp y \Leftrightarrow \langle x,y\rangle =0\) for any \( x, y \in \mathbb {H}\) ” plays the most fundamental role, without a shade of doubt. As we will see in this chapter, it is both useful and interesting to study an approximate version of orthogonality, in the more general setting of a Banach space \( \mathbb {X}.\) In an inner product space \( \mathbb {H} \) , a natural way to define approximate orthogonality is as follows: \(\begin{aligned}{}x\perp ^\epsilon y\Leftrightarrow |\langle x,y\rangle | \le \epsilon \Vert x\Vert \Vert y\Vert ,~\text {where~}x,y\in \mathbb {H}, \epsilon \in [0,1).{}\end{aligned}\) Due to the importance of B-J orthogonality in the study of geometry of operator spaces, the above notion has been generalized in Banach spaces.