Can one view $\ell_{1}^{4}$, $\mathbb{R}^{4}$ equipped with the $\ell_{1}$-norm, as a space of continuous functions on any extremally disconnected space?
Existence of extremally disconnected space $X$ for which $C(X)$ is $\ell_1^4$
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general-topology
functional-analysis
banach-spaces
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1Are this and [your other question](http://math.stackexchange.com/q/68983/) by any chance motivated by [this question](http://mathoverflow.net/questions/76179/) on MO? Then Bill Johnson already answered it in full... – 2011-10-01
1 Answers
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- Since the function space is 4-dimensional, the underlying space $X$ would have to consist of 4 points.
- The norm of $C(X)$ is the $\ell_\infty$ norm on $\mathbb R^4$.
- For $d>2$, endowing $\mathbb R^d$ with $\ell_1$ and $\ell_\infty$ norms results in non-isometric normed spaces. See How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?
(Based on t.b.'s comment)