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Can anybody please help me to prove this:

Let $p$ be greater than or equal to $1$.

Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p<\infty\}$ of all $p$-summable sequences (with norm $||u||_p=\sqrt[p]{\sum_{n=1}^\infty |u_n|^p}\ )$, there is an inner product $<\_\,|\,\_> $ s.t. $||u||^2=<u\,|\,u>$ if and only if $p=2$.

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    $\begingroup$ Hint: Parallelogramm law, $e_1$, $e_2$. $\endgroup$
    – martini
    Oct 18, 2012 at 13:10
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    $\begingroup$ Hint: Prove that if $p=2$, the parallelogram law is satisfied. WHen $p\neq 2$ give a counter example to the parallelogram law . $\endgroup$
    – Tomás
    Oct 18, 2012 at 13:11
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    $\begingroup$ Suppose $\ell_p$ is an Hilbert space. So its satisfies the paralelogram law. Write down the formula of the parallelogram law and conclude that the formula is only true for all vector if $p=2$. $\endgroup$
    – Tomás
    Oct 18, 2012 at 13:15
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    $\begingroup$ This question gives more details on parallelogram law: Norms Induced by Inner Products and the Parallelogram Law $\endgroup$ Oct 18, 2012 at 13:15
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    $\begingroup$ I think the question is whether $\ell_p$ admits an inner product making $\ell_p$ to a Hilbert space. The corresponding norm doesn't have to be $\Vert\cdot\Vert_p$ (an equivalent norm would do). None of the comments above seems to answer this question. $\endgroup$
    – user8268
    Oct 18, 2012 at 13:19

1 Answer 1

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Assuming we are working with the usual norm (as OP said in comments), suppose $\ell_{p}$ is an Hilbert space. So its must satisfy for all $u,v$: $$2\|u\|_{p}^2 + 2\|v\|_{p}^2 = \|u + v\|_{p}^2 + \|u - v\|_{p}^2.$$

As suggested by martini, take $u=e_{1}=(1,0,...,0,...)$ and $v=e_{2}=(0,1,0,...,0,...)$. Hence, by the last equality, we have $$4=2^{\frac{2}{p}}+2^{\frac{2}{p}}$$

Now you can solve the last inequality and verify that $p=2$.

On the other hand, if $p=2$, you can easily check that $\ell_{2}$ is a Hilbert space.

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