To show norm is arising from inner product

Question

Question is to show that $C([0,1])$ with respect to $\|\cdot\|_{1}$ is not inner product space i.e. there is no inner product agreeing with this norm. I think I have to show somehow parallelogram will not hold. But I am new to this topic and stuck how to do it. Any help. Thanks.

Answer 1

Take $f(x)=x$ and $g(x) =1-x$ and verify that $\|f+g\|^{2}+\|f-g\|^{2} \neq 2\|f\|^{2}+2\|g\|^{2}$. This implies that the norm is not given by any inner product.