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Let $L^{p}(\mathbb{R}^d)$ be the linear space consists of $L^p$-integrable functions on $\mathbb{R}^d$ for $1\le p \le \infty$. Are there any relation among these spaces?

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This must be a duplicate. However, the short answer is no. – Siminore Sep 11 '12 at 7:57
What do you mean by "relation"? What parameter are you varying ($p$ or $d$)? – Qiaochu Yuan Sep 11 '12 at 8:01
Maybe and answer your question? – t.b. Sep 11 '12 at 9:40

A simple observation, which apparently was not mentioned in either of the aforementioned threads.

If $p\le q\le r$, then $L^p\cap L^r\subset L^q\subset L^p+L^r$. This holds on any measure space, including $\mathbb R^d$.

Proof is based on the decomposition $f=f_S+f_L$ where $f_S=f\chi_{\{|f|\le 1\}}$ and $f_L=f\chi_{\{|f|>1\}}$. Indeed, $|f_S|^r\le |f_S|^q\le |f_S|^p$ and $|f_L|^p\le |f_L|^q\le |f_L|^r$ pointwise. Therefore, $f\in L^q$ implies that $f_S\in L^r$ and $f_L\in L^p$, proving the second inclusion. Proof of the first inclusion is left as an exercise.

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See also this Wikipedia entry for additional information. – Willie Wong Sep 12 '12 at 10:53

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