I need to prove that $L_q(\Omega) \subset L_p(\Omega)$ when $1\leq p \leq q \leq \infty$ when $\Omega$ is bounded. The hint given is that I should use Hölders inequality. How do I start my proof using this inequality?
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1Suppose $f\in L^q(\Omega)$. You want to see that $\int_{\Omega} \lvert f\rvert^p,d\mu$ is finite. Any idea how Hölder's inequality can give an upper bound on that integral? – Daniel Fischer Feb 28 '16 at 11:14
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already been addressed: http://math.stackexchange.com/questions/66029/lp-and-lq-space-inclusion – Jean Marie Feb 28 '16 at 11:54