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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.

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    It's hard to utilize the $p$-norm on arbitrary real valued functions, which may not be measurable or integrable..2011-10-19
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    @Ragib: I don't understand your point. The question seems to be: Consider the functions $f_p: \mathbb{R}^n \to \mathbb{R}$ given by $f_p(x) = \|x\|_p$ as elements of the space $X = C(\mathbb{R}^n, \mathbb{R})$ of continuous functions $\mathbb{R}^n \to \mathbb{R}$. Is there a norm on $X$ such that $f_{p} \to f$ with respect to that norm? I don't know (didn't think about it), but I would like to know what the motivation of this question is...2011-10-19
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    Of course, $\|x\|_{\infty} = \lim_{p \to \infty} \|x\|_p$ for all $x \in \mathbb{R}^n$ (even uniformly on compact subsets of $\mathbb{R}^n$). But why do you want to phrase this in terms of a **norm** on the space of continuous functions on $\mathbb{R}^n$?2011-10-20
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    @t.b.: Have you ever see a use for that identity (for $L^p$ norms for example)? I'm curious where that might turn out to be useful.2011-10-20
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    @JonasTeuwen it can be quite useful in PDE. For example, take the heat equation $u' = \Delta u$, $u(0) = f$. For $2 \leq p < \infty$ simple multiplier methods yield $\|u(t)\|_p \le \|f\|_p$. Taking limits the result holds for $p = \infty$ as well.2011-11-24

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