1
$\begingroup$

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?

  • 0
    No. The only possibility is a four point space. However, by looking at the combinatorial structure of the unit ball you'll see that $\ell^{1}(4)$ is not isometrically isomorphic to $\ell^{\infty}(4)$. Only $\ell^{1}(1)$ and $\ell^{1}(2)$ are isometrically isomorphic to the corresponding $\ell^{\infty}$-spaces, so only in dimensions $1$ and $2$ such a thing is possible.2011-10-01
  • 1
    Are 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 1