Proposition 2

The vector space V, equipped with the Zariski topology, is an irreducible noetherian space. In particular, every non-empty open subset of V is dense.

Since $A_V$ is noetherian, Spec(A$_V$) is noetherian (Commutative Algebra, Chap. II, §4, no. 3, Cor. 7 of Prop. 11), and every subspace of a noetherian space is noetherian (loc. cit., no. 2, Prop. 8). With the notation of the Cor. of Prop. 1, the intersection of the $\mathfrak{m}_x$ is $\{0\}$, and $\{0\}$ is a prime ideal of $A_V$; thus V is irreducible (loc. cit., no. 3, Prop. 14).