0

Suppose $1<p,q<\infty$, $\dfrac{1}{p} + \dfrac{1}{q}=1$, $(X,\Sigma,\mu$ is $\sigma$-finite measure space and $g$ is a measurable function such that $fg \in L^1(X)$ for every $f \in L^p(X)$. Prove that $g \in L^q(X)$.

I have no idea where to start this question. It seems to me a lot like a close question to the Riesz Representation Theorem, or an application of it. Any help is much appreciated. I'm just refreshing many questions I haven't looked at in quite some time for preparation of my Analysis Qualifying Exams.

Alex
  • 243
  • Are you allowed to use that the dual of $L^p$ is $L^q$? (Maybe this is what you cal the "Riesz representation theorem"?). – Etienne Jul 27 '13 at 22:41
  • Thank you for pointing me in the right direction. I am finding that I should change variable names a few times to try and find my problem elsewhere. Sorry for my second duplication of the day. – Alex Jul 28 '13 at 03:12
  • Don't worry for the duplications: I guess it can happen any time! – Etienne Jul 28 '13 at 04:18

0 Answers0