Prove there cannot be an inner product which turns $l^p$ into an inner product space?

Question

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be exist an inner product $\langle\cdot , \cdot\rangle:l^p \times l^p \rightarrow \mathbb{C}$ for which $\langle x,x\rangle=\|x\|_p^2$ for each $x=(a_k)_{k\geq 1}$.

Thanks, any help will be appreciated, I am completely stuck.

Answer 1

Let $p$ be such that there is an inner product for which $\langle x,x\rangle=\lVert x\rVert^2_p$. Using the parallelogram identity $$\langle x+y,x+y\rangle+\langle x-y,x-y\rangle=2\langle x,x\rangle+2\langle y,y\rangle$$ with $x=(1,0,0,\dots)$ and $y:=(0,1,0,\dots)$, we get that $2^{2/p}=2$, hence $p=2$.