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Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by

$$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$

with $W$ the LambertW function defined by $W(x e^x)=x$ for all $x\geq 0$.

Define the metric $d(x,y)=\rho(x-y)$ for $x,y\in\mathbb{R}^n$.

Question: Is the metric space $(X,d)$ normable?

My attempt: Since a topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of 0, and since all metric spaces are Haussdorf, it suffices to show that it has a convex bounded neighborhood of 0, which it has, since its unit ball is convex. Is this correct?

Carucel
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    What's the motivation of this question? –  Jul 24 '16 at 18:18
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    @ArcticChar, I am trying to make a modification to the Euclidean norm defined by $\left|x\right|\cdot\left|x\right|=\sum\left|x_i\right|\cdot\left|x_i\right|$, by replacing $+$ with $\cdot$ and $\cdot$ with $\wedge$ (think about the Ackermann function). Doing so yields $\left|x\right|=\frac{\log(\prod x_i^{x_i})}{W(\log(\prod x_i^{x_i}))}$, but this is not a norm, so I added the logarithm and replaced $x$ by $e^x$, to obtain the form in the question. – Carucel Jul 24 '16 at 18:20
  • The answer to "this metric induced by a norm?" is no. The answer to "is there a norm that gives the same topology as this metric?" is yes. It is not clear which question you meant. –  Jul 24 '16 at 18:46
  • @ᴡᴏʀᴅs, thank you: the title was incorrect, I updated it. – Carucel Jul 24 '16 at 18:56
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    The formula for $\rho$ doesn't matter: any $n$-dimensional Hausdorff TVS is just $\mathbb{R}^n$. How to endow topology on a finite dimensional topological vector space? –  Jul 24 '16 at 19:12
  • @ᴡᴏʀᴅs, thank you! This is quite a surpise for me... Does this also mean that if $\rho$ is defined on $\mathbb{R}^\mathbb{N}$, instead of $\mathbb{R}^n$, the topology is the same as the topology induced by the $\ell^2$ norm? – Carucel Jul 24 '16 at 20:57
  • No, finite-dimensional matters. All the different $\ell^p$ norms induce different topologies in infinite dimensions. –  Jul 24 '16 at 21:00
  • Let us continue this discussion in chat. – Carucel Jul 24 '16 at 21:06

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