This may be a silly question but every two norms on $\mathbb{R}^n$ are equivalent and $\Vert\cdot\Vert_2$ comes from the usual dot product so $(\mathbb{R}^n,\Vert\cdot\Vert_2)$ is (at least) an inner product space (pre-Hilbert).
Why can't we say that $(\mathbb{R}^n,\Vert\cdot\Vert_{\infty})$ is also pre-Hilbert? I know the uniform norm doesn't come from an inner product but two spaces are (topologically) the same if their norms are equivalent and up until now (at least in my course) it is only the topology that gives the space its "uniqueness".
The fact that a norm comes from an inner product doesn't change the topology nor the algebraic (vector space) structure. I get the product may be useful but what unique, defining, core property ... does it have??