Proposition 1

Let H be the set of algebra homomorphisms from $A_V$ to $k$. For any $x\in V$, let $h_x$ be the homomorphism $f \rightarrow f(x)$ from $A_V$ to $k$. Then, the map $x \rightarrow h_x$ is a bijection from V to H.

Indeed, let $H'$ be the set of algebra homomorphisms from $k[X_1, . . . ,X_n]$ to $k$. The map $\chi \rightarrow (\chi (X_1), . . . , \chi (X_n))$ is clearly a bijection from $H'$ to $k^n$.