maximum norm in an inner product space

Question

the question:

Let $V$ be the vector space of all continuous functions over the interval $[0,1]$. Does an inner product space over $V$ that define this norm: $|f|_{\infty}=max_{0\le x\le1}|f(x)|$ exist? and if so can you "give up" the continuity?

my approach

I tried to solve this one but I don't exactly know what this norm: $||_{\infty}$ means so please explain this first. I figured out that if such an inner product space exist you cannot "give up" the continuity from calculus.

Answer 1

$\Vert f\Vert_{\infty}$ is just the largest value that $|f|$ takes on the interval $[0,1]$, which has to be finite by continuity.

There is no inner product that induces this norm because it fails to satisfy the parallelogram law: $$\Vert f+g\Vert^2 +\Vert f-g\Vert^2 =2\Vert f\Vert^2 +2\Vert g\Vert^2$$ for all continuous $f$ and $g$ on $[0,1]$. Every inner product-induced norm satisfies the parallelogram law. Try to find $f$ and $g$ where it does not hold.