Does the $p$-norm on $\mathbf{R}^n$ converge to the max-norm on $\mathbf{R}^n$ as elements in the space of real valued continuous functions on $\mathbf{R}^n$ endowed with some norm?
More precisely, consider $V = C(\mathbf{R}^n, \mathbf{R})$. Does there exist a norm $\Vert \cdot \Vert$ on $V$ such that the sequence $(\Vert \cdot \Vert_p)_p$ converges to the maximum norm $\Vert \cdot \Vert_\infty$ with respect to $\Vert \cdot \Vert$?
Here's the motivation for this question.
In some sense, I though the max-norm should be the limit of the $p$-norms as $p$ goes to infinity. "Taking an $\infty$-th root of the sum of the infinite powers" in some sense should be the maximum norm. I just thought that this could be made precise.