The following definitions are from Mumford's book on Abelian Varieties:
A group scheme $G$ is a finite type scheme over an algebraically closed field $k$ with morphisms $m:G\times G\rightarrow G$, $e:\operatorname{Spec}(k):\rightarrow G$ and $i:G\rightarrow G$ such that for any finite type scheme $S$ over k, the maps $m$, $e$ and $i$ induce a group structure on $G(S)= Hom_{\operatorname{Spec}(k)}(S, G)$.
For any closed $x\in G$ we define $$\textbf{H}_x= Hom_{cont}(\mathcal{O}_{x,G}, k)$$ Now, the hyperalbegra $\textbf{H}$ of $G$ is defined as: $$\textbf{H}=\bigoplus_{\operatorname{closed}x\in G}\textbf{H}_x$$ where $Hom_{cont}$ is the are the maps $L:\mathcal{O}_{x,G}\rightarrow k$ which are continuous in the sense that $L(m_x^{n}) = (0)$ for some natural number $n$ (i.e. continuous in the respective m-adic topologies).
My question is the following. What sort of continuous maps are being considered? Are they morphisms between vector spaces or morphism between groups?
I believe the answer is vector spaces because the following is stated immediately after:
$$\textbf{H} = \varinjlim_{Z\in \Sigma}\Gamma(\mathcal{O}_Z)^*$$ where $\Sigma$ is the collection of $0$-dimensional subschemes of $G$ and $W^*$ is the dual of any $k$-vector space $W$.
If my guess is incorrect, could someone please help me with the proof of the equality stated above?
Edit: I believe I see why my question might now be clear enough! My doubts would be answered if someone could help prove the equality mentioned above: $$\textbf{H} = \varinjlim_{Z\in\Sigma}\Gamma(\mathcal{O}_Z)^*$$
Thank you!