I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the common solutions from sets like fixed points, equilibrium problem solution sets, variational analysis etc. Let me introduce it for a smooth Banach space:
Let $X$ be a smooth Banach space, where the duality mapping $J$ is single-valued, and we define $V: X \times X \to \mathbb{R}$ by $V(x, y)=\Vert x \Vert^2 - 2 \langle x, Jy \rangle + \Vert y \Vert^2$, for $x, y \in X$. This does not behave like a metric in general Banach spaces. I am not understanding the fact that why it has been employed in Banach spaces, whereas we can always find a metric associated with a Banach space anyway. Can anyone please help me understanding this. Thank you.