Are there any examples of parabolic PDEs $$u' - Au = f$$ posed in a Gelfand triple setting $V \subset H \subset V^*$ with $V$ and $H$ chosen NOT as spaces of functions (or distributions) over a domain $\Omega \subset \mathbb{R}^n$ or a manifold $M$? Here $A\colon V \to V^*$ is some elliptic operator.
I'm looking for some applications of this to some weirder/more interesting contexts than the usual Lebesgue or Sobolev spaces. Of course I know theory of existence etc can be done abstractly in Banach spaces, but I want applications. Thanks.
