Let $\Omega \subset \mathbb{R}^n$. Consider, for example, the Sobolev space $H^1_0(\Omega)$. It is known that the dual is $H^{-1}(\Omega)$ and is Banach with respect to the operator norm
$$||f||_{-1} = \sup \limits_{v \in H^1_0} \dfrac{(f,v)}{||v||_1}. $$
Where $(\cdot, \cdot)$ is the duality product, which is the $L^2$ inner product.
Can we define another duality product that can still be used to define the operator norm?
My question originates from trying to understand what is a duality product, to begin with, and why are we using the $L^2$ inner product for that. I encountered this in the study of PDEs.
In general, What if, for example, we use the $H^1$ inner product? Or what if we are looking for the duality product of a different space? How would one define the duality product?