# Classical theorem of Riesz and Banach

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?

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I don't know the name of the theorem (I didn't even know it had a name or was a theorem), but it's an easy consequence of the Hahn-Banach theorem and of the $L^p - L^q$ duality, sometimes called Riesz theorem. –  nik Mar 17 '12 at 14:31
It's one of the "important consequences" listed in this link for the Hahn-Banach Theorem. –  David Mitra Mar 17 '12 at 14:32
See here also; I believe this is why Riesz is mentioned. –  David Mitra Mar 17 '12 at 14:39
Thanks for the comments! –  Roupam Ghosh Mar 17 '12 at 14:55
I would think this comes down to two theorems: It is a consequence of the Hahn–Banach theorem that any proper, closed subspace of a Banach space is contained in the null space of a non-zero continuous functional. And the Riesz representation theorem states that the dual of $L^p$ is $L^q$ (if $1\le p<\infty$ and $1/p+1/q=1$).