Are $l^p$ and $L^{p}$ space Hilbert spaces?Except for $p=2$,I can not find a standard inner product on them.
My opinion:$l^p$ and $L^p$ spaces are Hilbert space iff $p=2$(Not sure,an intuition without a reliable proof).
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Read about the parallelogram law. – Aweygan Jun 07 '17 at 06:30
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Thank you for pointing out straightly. – Muses_China Jun 07 '17 at 06:41
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A Banachspace $(X,\|\cdot\|)$ is a Hilbertspace if and only if the norm satisfies the parallelogram law.
The accepted example here proves for $1\leq p\leq\infty$ that $L^p([0,1])$ is a Hilbertspace if and only if $p=2$.
Mundron Schmidt
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I don't know Daniel Schepler, but from the link you can see, that if the space is a Hilbertspace, then we get $\langle u,v\rangle=\frac14(|u+v|^2-|u-v|^2)$ which is bilinear if and only if $p=2$. – Mundron Schmidt Jun 07 '17 at 06:47