I'm looking for an example of a normed space in which the norm doesn't create an inner product. I know that the condition for that is that the Parallelogram Law doesn't apply.
I tried using some $L^p$ spaces, but I soon discovered that there is in fact an inner product. The other examples that came to my mind were in $\Bbb R^n$, but there I'm sure that the norm always induces an inner product.
$C[0,1]$ is a good example. Consider $f$ and $g$ with the following properties: $0\leq f,g \leq 1$, $f(x)=0$ for $x \leq \frac 12 -\frac 1 n$, $f(x) =1$ for $x \geq \frac 1 2$, $g(x)=1$ for $x \leq \frac 12 -\frac 1 n$, $g(x) =0$ for $x \geq \frac 1 2+\frac 1 n$. Show that $\|f+g\|^{2} +\|f-g\|^{2} <2\|f\|^{2}+2\|g\|^{2}$.