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.
Best Answer
As t.b. noted, we are looking at the functions $f_p:{\mathbb R}^n\to {\mathbb R}$ given by $f_p(x)=\|x\|_p$ and want to know if they converge to $f_\infty$ in some norm on $C({\mathbb R}^n,{\mathbb R})$. It is easy to see that $f_p \to f_\infty$ uniformly on compact sets, and so if you consider the space $C(\Omega,{\mathbb R})$ where $\Omega \subset {\mathbb R}^n$ is bounded and open, then $f_p \to f_\infty$ in, for example, all $L^p$ norms on $C(\Omega,{\mathbb R})$.
For $C({\mathbb R}^n,{\mathbb R})$, the problem is that the the functions $f_p$ do not belong to $C({\mathbb R}^n,{\mathbb R})$, when you endow it with any of the standard norms, such as $L^p$ norms, so we cannot even talk about convergence to $f_\infty$. That being said, you can look at less common norms such as
$\|f\|_X = \sup_{r >0}\left( e^{-r}\sup_{|x|\leq r} |f(x)|\right)$
or weighted $L^P$ norms
$\displaystyle \|f\|_{w,p} = \left( \int_{{\mathbb R}^n} w(x) |f(x)|^p dx\right)^{1/p}$
where $w$ is positive weighting function which decays to zero (exponentially) as $|x|\to \infty$ in order to "cancel out" the growth of $f$. Then the sequence $f_p$ would actually belong to the space $C({\mathbb R}^n,{\mathbb R})$ when endowed with either of these norms. I haven't written this down, but it looks like it would be very easy to show that $\|\cdot\|_p \to \|\cdot\|_\infty$ in both $\|\cdot\|_X$ and $\|\cdot\|_{w,p}$