Show that the sup-norm is not derived from an inner product I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$  does not come from an inner product (the norm is defined on all bounded and continuous real valued functions).
I tried to show that the inner product does not hold by using the conjugate symmetry, linearity and non-degenerancy conditions. But I am unsure of how to do it for the norm $\lVert{\cdot} \rVert _{\infty}$


$\|u+v\|^2+\|u−v\|^2=2\|u\|^2+2\|v\|^2$

the parallelogram identity
the parallelogram identity is necessary and sufficient for a given norm to be derived from an inner product.
You can show that in your case it does not hold.
