TITLE: $\ell_p$ is Hilbert if and only if $p=2$
QUESTION [28 upvotes]: 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 $p=2$.

REPLY [35 votes]: 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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