lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -164,12 +164,16 @@ entire \(Q\)-coset it enhabits -- i.e.
 \(\operatorname{supp}_{\operatorname{ess}} K[x, x^{-1}] = 2 \mathbb{Z}\). This
 isn't always the case. Nevertheless, in general we find\dots
 
-% TODOO: Explain what we mean by Zariski topology in 𝔥*
 \begin{proposition}
   Let \(V\) be an infinite-dimensional admissible representation of
   \(\mathfrak{g}\). The essential support
-  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense in
-  \(\mathfrak{h}^*\).
+  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense\footnote{Any
+  choice of basis for $\mathfrak{h}^*$ induces a $K$-linear isomorphism
+  $\mathfrak{h}^* \isoto K^n$. In particular, a choice of basis induces a
+  unique topology in $\mathfrak{h}^*$ such that the map $\mathfrak{h}^* \to
+  K^n$ is a homeomorphism onto $K^n$ with the Zariski topology. Any two basis
+  induce the same topology in $\mathfrak{h}^*$, which we call \emph{the Zariski
+  topology of $\mathfrak{h}^*$}.} in \(\mathfrak{h}^*\).
 \end{proposition}
 
 This proof was deemed too technical to be included in here, but see proposition