Consider $$X:=\bigcap_{1\le p<+\infty} L^p(\mathbb{R}).$$
Due to the interpolation property, one can easily write $$X=\bigcap_{n\in\mathbb{N}} L^n(\mathbb{R}),$$ so it is not difficoult to see $X$ as a Frechet space (take, for instance, the tha family of seminormos $\||\cdot\||_{L^n(\mathbb{R})}$).
Does anyone know a reference where I the (Frechet) topology of $X$ is studied?
NOTE: I've found such reference for $L^p([0,1])$, that is, for a finite measure space. It is important to get the reference for, at least, a $\sigma$-finite set like $\mathbb{R}$ (withe Lebesgeue measure).