The following is the Kac denominator formula:
$\prod_{\alpha \in \Phi^+} (1 - e_{-\alpha})^{m_\alpha} = \sum_{w \in W} \varepsilon (w) e_{w(\rho) - \rho}$
(Where the equality is in the ring $\mathfrak{R}$ of functions on $H^*$)
For Kac-Moody algebras, both the set of positive roots $\Phi^+$ and the Weyl group $W$ are infinite sets. Each individual term of the product on the LHS (and the sum on the RHS) are also elements of $\mathfrak{R}$. So it seems like there is infinite product of ring elements in the LHS (and an infinite sum of ring elements in the RHS). I want to understand how this makes sense. (Especially given that there is no notion of convergence here, since the entire treatment is purely algebraic (right?))