Why is the weight lattice a lattice?

Question

For $\mathfrak g$ a complex semisimple Lie algebra, $\mathfrak h \subset \mathfrak g$ a CSA and $R$ the corresponding set of roots we call the following set $\Lambda_w=\{\lambda \in \mathfrak h^*|\langle\lambda, \alpha^\vee\rangle\in \mathbb Z \text{ for all } \alpha\in R\}$ the weight lattice. Why is that set a lattice? It surely is an additive subgroup of $\mathfrak h^*$ and it spans $\mathfrak h^*$ over $\mathbb C$ (since roots are contained in the weight lattice). However, why is it isomorphic to $\mathbb Z^n$ for $n$ the dimension of $\mathfrak h$? Why do the fundamental dominant weights form a $\mathbb Z$-basis of the weight lattice?