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I would like to know why $W^{k,2} (\Omega) $ is a Hilbert space , why is it impossible to define inner product in other Sobolev spaces, ie exponent $\ge2$ . Here $||u||_{W^{k,2} (\Omega)} $ = $(\sum_{|\alpha|\le k}||D^\alpha u||^2_{L^2(\Omega)})^{1/2}$. I would want to know it technically as well.

Rudy the Reindeer
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Theorem
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  • I'd say for the same reason why $L^p(\Omega)$ spaces are Hilbertian only for $p=2$. After all, $W^{k,p}(\Omega)$ is just a closed subspace of $L^p(\Omega)\times L^p(\Omega) \times ... \times L^p(\Omega)$. – Giuseppe Negro May 24 '12 at 11:08
  • I don't think that statement is very easy to prove. Think about the case $L^p(\Omega)$ first which looks much more trivial than it actually is. @Giuseppe: and how do you prove that? Of course the usual norms are not Hilbert norms as they fail the parallelogram law but you'd need to prove that there are no equivalent norms making them into Hilbert spaces. – t.b. May 24 '12 at 11:10
  • @t.b. No, I did not understand the question as you did. Ananda fixed the norm. – Siminore May 24 '12 at 11:11
  • @t.b.: Heh, I'd leave that as an exercise for the reader. :-D – Giuseppe Negro May 24 '12 at 11:17

1 Answers1

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As $1\leq p \leq \infty$, the only Hilbert space among $L^p$ spaces is $L^2$. You can use the parallelogram law to prove this. See here.

Edit: A Banach space $X$ admits an equivalent norm $\|\cdot \|$ such that $\| \cdot \|^2$ is twice Fréchet differentiable on $X$ if, and only if, $X$ is isomorphic to a Hilbert space. This is stated here. But, as I wrote in the comments, Sobolev spaces are interesting when they are endowed with their natural norms.

Siminore
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  • Again: Ananda fixed the norm of the Sobolev space. Or, better: his/her question is rather confused, since he/she writes "impossible to define" but then writes down the standard norm. What is the question? By the way, Sobolev spaces are useful as long as they are endowed with their standard norms. – Siminore May 24 '12 at 11:14
  • @t.b my question is exactly what u are talking about. – Theorem May 24 '12 at 11:26
  • @Siminore Thanks. – Theorem May 24 '12 at 11:30
  • Okay, thanks for the edit, +1. I removed my comments as they are no longer relevant. An alternative approach and links (see especially the answer on MO by Bill Johnson) is given in this answer. – t.b. May 24 '12 at 11:43