It is well known that $(H^1)^*=BMO$ (mod some identifications). The way this is usually proved is by proving that every continuous linear functional on a suitable dense subspace $X$ of $H^1$ can be represented via the pairing defined on $X\times BMO, \langle f,g\rangle=\int fg$. It is indeed necessary to use a subspace since one can construct explicit pairs $(f,g)$ with $f$ in $H^1$ and $g$ in $BMO$ such that the integral is not absolutely convergent. I seem to recall reading somewhere about a functional analysis proof of the impossibility of defining such an integral pairing between the two spaces, but I can't seem to manage to track it down or reconstruct it.
Does anyone know of such a proof? Any reference or idea is welcome.