coprime weight vector ${\underline}{p}$ such that ${\underline}{\ell}={\underline}{p}^{\,\sharp}$.

Let ${\underline}{p}$ be pairwise coprime and ${\underline}{\ell}={\underline}{p}^{\,\sharp}$.

For any prime $m$, ${\mathcal}{M}_k(m)$ is admissible if and only if $m$ does not divide $\ell_k=\prod_{i\neq k}p_i$. It follows that $\gcd(m,p_i)=1$ for all $ i\neq k$. The effect of ${\mathcal}{M}_k(m)$ is multiplying $p_k$ by $m$. Note that the new weight vector has still the form ${\underline}{\ell}'={\underline}{p}'^{\,\sharp}$ with ${\underline}{p}'$ pairwise coprime.

The division ${\mathcal}{D}_k(m)$ is admissible if and only if $m$ divides $\ell_j=\prod_{i\neq j}p_i$ for all $j\neq k$ and does not divide $\ell_k$. It follows that $\gcd(m,p_i)=1$ for all $i\neq k$, and $m$ divides $p_k$. The effect of ${\mathcal}{D}_k(m)$ is to divide $p_k$ by $m$. The new weight vector has still the form ${\underline}{\ell}'={\underline}{p}'^{\,\sharp}$ with ${\underline}{p}'$ pairwise coprime. Thus, any iterated application of admissible multiplications or divisions transforms ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ into ${\underline}{\ell}'={\underline}{p}'^{\,\sharp}$, with ${\underline}{p}'$ pairwise coprime. Starting with ${\underline}{p}={\underline}{\ell}=(1,\ldots,1)$ this proves the implication “$\,\Rightarrow$”.

*Proof of “$\,\Leftarrow$”.* Let ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ with ${\underline}{p}$ pairwise coprime. From the discussion above, ${\mathcal}{D}_k(m)$ is admissible for any prime factor $m$ of $p_k$. By repeated application of admissible divisions, we can transform ${\underline}{p}$ into $(1,\ldots,1)$, and this proves that ${\mathbb{P}}({\underline}{\ell})\simeq{\mathbb{C}\mathrm{P}}^n$.

Quantum spaces ${\mathbb{P}}_q({\underline}{\ell})$ with weight vectors as in Theorem \[thm:6\] form a nice class 