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Are all the norms of $L^p$ space equivalent? That is, for any $p,~q \in R^+$, there exist two positive number $C_1,~C_2$ such that $$ C_1\|u\|_{L^q} \le \|u\|_{L^p} \le C_2\|u\|_{L^q}. $$

chentaocuc
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    No. Consider $\frac{1}{(1+\lvert x\rvert)^{n/p}}$. It's in $L^q(\mathbb{R}^n)$ for $q > p$, but not in $L^p(\mathbb{R}^n)$. – Daniel Fischer Jul 28 '13 at 13:03

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If fact for every $p$ and $q$ there is a function $f$ with $\|f\|_p<\infty$ and $\|f\|_q=\infty$. Using this you can show that for any measure space $X$ (I'm assume your questions is about $L^p(\mathbb{R})$) if there is a $p$ and $q$ with $\|\cdot\|_p$ equivalent to $\|\cdot\|_q$ then they are all equivalent and all the $L^p(X)$ are finite dimensional.

Owen Sizemore
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