In the following exercise:
Use the parallelogram equality to show that the Sup-norm does not arise as an inner product.
there's no information about the vector space, so I assume it's applied for any vector space. For some specific vector space e.g $\mathit{C}([0, 1])$, $\mathscr{l}^1$, $\mathscr{l}^{\infty}$, I can show that it's true by constructing counter-examples for the parallelogram equality.
However, for arbitrary vector spaces, it seems hard to reconcile all the definitions of the sup-norm for various vector spaces to proceed. Is there any way to get around with this issue?
Case 1: $C[0,1]$ is a normed space with the sup norm $\displaystyle\|f\|_\infty = \sup_{x\in [0,1]}|f(x)|$.
Case 2: $\ell^\infty$ is a normed space with the sup norm $\displaystyle\|x\|_\infty = \sup_{n\in \mathbb{N}}|x_n|$.
In the both cases, you should calculate, separately, $\|x+y\|^2_\infty +\|x-y\|^2_\infty$ and $2\left(\|x\|^2_\infty+\|y\|^2_\infty\right)$. Thus visualizing that in both cases, the sup norm does not arise as an inner product.