The complete question goes like: Suppose V is a closed codimension 1 subspace of $L^p(\mu)$ for $p\in (0,\infty)$. Show that there is a $g\in L^{p’}(\mu)$ such that $f\in V$ if and only if $\int fg d\mu=0$.
I wanted to build a linear functional vanishing on V, but don’t know how to start. I think it has something to do with the Rietz representation theorem, but still I don’t see how. Thank in advance!
Let $f_0 \in L^{p} (\mu) \setminus V$. In general, the argument in Hahn Banach Theorem show that $\Lambda (x+af_0)=a,x \in V,a \in \mathbb R$ defines a continuous linear functional on $ L^{p} (\mu)$. When V has codimension 1, $\{x+af_0:x \in V,a \in \mathbb R\}=L^{p} (\mu)$. This linear functional is given bay an element of the dual $L^{p'} (\mu)$. All you have to observe is $\Lambda (x+af_0)=a=0$ if and only if $x+af_0 \in V$.