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I'm studying $L^p$,$l^p$ spaces recently, but I don't see the motivation for this. The only application I know is that $l^2$ can be used to characterize Hilbert spaces up to dimension. Is $L^p$ spaces just for recreational mathematics.

I also don't see why $l^p$ is invented. Moreover, I know that $l^2(\mathbb{N})$ and $L^2(\mathbb{R}^n)$ are isometrically isomorphic, but I'm curious when $l^2$ is much useful to analyze and when $L^2$ is much useful to analyze the structure.

Thank you in advance.

Rubertos
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  • $\ell^p$ (sequence spaces) and $L^p$ (Lebesgue spaces) are different. Which are you talking about? – Ian Apr 29 '16 at 16:42
  • @Ian I meant $L^p$ by an abstract measure so that $l^p$ is a type of $L^p$. And moreover, I don't even see a motivation for the special case $L^p(\mathbb{R}^n)$. – Rubertos Apr 29 '16 at 16:43
  • Related: http://math.stackexchange.com/questions/843108/why-are-lp-spaces-so-ubiquitous?rq=1 – yoyostein Apr 29 '16 at 16:47
  • $L^1,L^2$ and $L^\infty$ come up all the time. Some other $L^p$ spaces come up as the space being Sobolev embedded into by $W^{k,q}$ where $q$ is $1$ or $2$. These Sobolev spaces come up naturally in PDE theory. – Ian Apr 29 '16 at 16:48
  • Also, $L^2$-spaces naturally pop up in quantum mechanics, you can google Hilbert spaces in quantum mechanics to see how they are studied. – Mathematician 42 Apr 29 '16 at 17:04
  • For example, $W^{1,2}(D)$, where $D$ is any Lipschitz domain in $\mathbb{R}^3$, is the natural setting for linear PDEs related to the Laplacian in physical space. This embeds into $L^q$ for $q \leq 6$, and the embedding is compact when $q<6$. This is occasionally useful in PDE theory, for example in deducing the Poincare inequality. – Ian Apr 29 '16 at 17:07

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