I am new to functional analysis. So maybe I am missing something obvious.
I was reading this paper A closure problem related to the Riemann zeta function -- by Arne Beurling http://ukpmc.ac.uk/ukpmc/ncbi/articles/PMC528084/pdf/pnas00720-0060.pdf
Just before equation (5) he mentions,
If $C$ is not dense in $L^p$, we know by a classical theorem of F. Riesz and Banach that the dual space $L^q$ must contain a nontrivial element $g$ which is orthogonal to $C$ in the sense that $$ 0 = \int_0^1 g(x)f(x) dx \quad \quad f\in C$$
I could not see any references in the paper either where the classical theorem is mentioned, and so I do not have the slightest clue about that.
Can someone please point to some references or the name of the theorem?