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For $p<q$, using Holder inequality, I deduced that $$ \int_{[0,1]}|f|^p dx = \int_{[0,1]}|f|^p\cdot 1dx \leqslant\left(\int_{[0,1]}|f|^{p\cdot {q\over p} } dx\right)^{p\over q} \left(\int_{[0,1]}1^{ q\over q-p } dx\right)^{q-p\over q} =\left(\int_{[0,1]}|f|^q dx\right)^{p\over q}, $$ thus, $\Vert f\Vert_p\leqslant\Vert f\Vert_q$, which means $\Vert\cdot\Vert_p$ is weaker than $\Vert\cdot\Vert_q$.

I wish to figure out (1) Whether my deduction was correct? (2) How to further show that these two norms are not equivalent? Note that the space is $C[0,1]$.

Anne Bauval
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Max Young
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Hint:- Do you know that $C[0,1]$ is dense in $L^{p}$ ? . Now do you see the problem?. $L^{p}$ is not equal to $L^{q}$ if $p\neq q $. But if norms are equivalent then what can you conclude using the density of $C[0,1]$?

Mr. Gandalf Sauron
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