In class I was asked to show that there is no inner product on $\ell^1(\mathbb{N})$ which gives rise to the norm $\|\cdot\|_1$. I was able to do so, using the parallelogram law.
Now, I am wondering if it is possible for a norm on $\ell^1(\mathbb{N})$ which is equivalent to $\|\cdot\|_1$ to be induced by an inner product. It isn't immediately obvious to me whether or not this could be the case.
So far, I have tried using the definition of equivalent norms to no avail.