In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$. I am interested to understand the structure we can put on intersection of $L^p$-spaces.

For $I$ an interval of $[1,+\infty]$, we define $$L^I = \bigcap_{p\in I} L^p.$$ Obviously, because $L^a \cap L^b \subset L^c$ for $1\leq a\leq c\leq b \leq \infty$, we have that $L^{[a,b]} = L^a \cap L^b$. Thus, $L^I$ is a Banach space for the norm $\lVert \cdot \lVert_a + \lVert \cdot \lVert_b$.

**Questions:** What can we say of the structure of $L^I$ if $I$ is an open (or semi-open) interval of $[1,\infty]$? What is the more natural topology on it that make it complete? Are these spaces studied somewhere, for instance as a subpart of a more general theory?

Extensions:

I am also interested in spaces of the form $$L^p_+ = \bigcup_{\epsilon >0} \bigcap_{0<r<\epsilon} L^{r+p},$$ with similar questions as previously.

What happens if $I \subset (0,\infty]$, knowing that $L^p$-spaces for $p<1$ are quasi-Banach spaces?

Bull. Aust. Math. Soc.,85(2012) 433–445, doi:10.1017/S0004972711003510. $\endgroup$