Suppose $1<p,q<\infty$ are conugate indices, and $f\notin L^p(X,\mu)$. Show that the set $A=\{g\in L^q(X,\mu): fg\in L(X,\mu) \text{and} \int_X fg d\mu=0\}$ is dense in $L^q(X,\mu)$.
My Work:
Define $L_f: L^q\rightarrow$ scalar field where $L_f(g)=\int_X fg d\mu$. Then ker $L_f=A$. So I can use the result of my previous post Let $X$ be a linear normed space, and $L$ a nontrivial linear functional on $X$. Prove the following are equivalent:.
So, I tried to show $L_f$ is not bounded. To the contrary assume not. Hence there is $c>0$ such that given $g$, $|\int_X fg d\mu|\leq c||g||_q$. How can I proceed now?
