I was reading the Wikipedia article about weak topology and I was a bit confused by the notion of pairing. It seems like a more general way to put two spaces in duality but I was wondering if there are cases where it is interesting to consider a different pairing than \begin{align} X\times X^*&\to \mathbb{F}\\ (x,f)&\mapsto f(x) \end{align}
(or the scalar product which is the "same thing" by Riesz representation)
recall : Given two vector spaces $X, Y$ , a duality pairing between $X$ and $Y$ is a bilinear map $_X\langle·, ·\rangle_Y : X × Y → \mathbb{R}$ with the following properties:
(i) for all $x ∈ X\setminus\{0\}$ there exists $y ∈ Y \setminus \{0\}$ such that $_X\langle x, y\rangle_Y = 0$;
(ii) for all $y ∈ Y \setminus \{0\}$ there exists $x ∈ X \setminus \{0\}$ such that $_X\langle x, y\rangle_Y = 0$.