TITLE: A Banach space that is not a Hilbert space
QUESTION [12 upvotes]: Can someone give me an example of a Banach space that is not a Hilbert space? I can't think of any because I don't know how to show one space that can not have inner product structure.

REPLY [8 votes]: If a Banach space $(X, \|\cdot\|$ is a Hilbert space, then the norm satisfies the "parallelogram identity"
$$
\|x+y\|^2+\|x-y\|^2=2\big(\|x\|^2+\|y\|^2\big).
$$ 
But the norm of $C[0,1]$ does not satisfy such an identity: for $f=1$ and $g=x$,
$$
\|f\|=1,\,\,\|g\|=1,\,\,\|f+g\|=2,\,\,\|f-g\|=1,
$$
and hence
$$
5=\|f+g\|^2+\|f-g\|^2\ne2\big(\|f\|^2+\|g\|^2\big)=4.
$$