Does $L^1(R)\cap L^2(R)$ have finite or infinite corank in $L^2(R)$?
I guess the latter is the case but I have never seen this discussed, and would like to see a simple proof.



A question regarding $L^1(R)\cap L^2(R)$
It is a truth universally acknowledged, that a Banach space in possession of a continuous and dense inclusion in a strictly larger Banach space must have infinite co-dimension in it. This is also the case of 
 $$L^1(\mathbb{R})\cap L^2(\mathbb{R}),\ \|\cdot\|\_1+\|\cdot\|\_2 \longrightarrow L^1(\mathbb{R}) ,\ \|\cdot\|\_1 . $$ 


