Let $E$ be banach and $J:E \to E*$ be the duality map, that is, $<J(x),x>= \lvert \lvert x \rvert \rvert ^2$ and $\lvert \lvert J(x) \rvert \rvert = \lvert \lvert x \rvert \rvert$. Note that J does not have to be linear.
Proposition: If J were linear, then E is actually Hilbert.
I went ahead and defined $<x,y>:=<J(x),y>$ and it holds all I need, except symmetry (I'm working over the reals). I can't show that $J(x)(y)=J(y)(x)$ and can't see a reason why it should. I actually had a different proof, using parallelogram law, but my prof told me I was complicating myself and hinted me with the definition above. Any tips? Is there something obvious that I'm not seeing?
Define inner product as $ \langle x, y \rangle =\frac {J(x)y+J(y)x} 2$. This should resolve the issue with symmetry without affecting other properties.