Bijection between a variety $X$ and $\hom(k[X], k)$
I found the above question is very interesting. Despite the illuminating answer provided in the link. How could we use Nullstellensatz to prove surjectivity?
I believe that $k[X]$ should be the coordinate algebra for some variety $X$.
Let $\psi$ be a $k$-linear homomorphism. Let $\psi(x_i) = p_i$ and $p = (p_1,...,p_n) \in \Bbb A^n$. Then $\psi$ agrees with $ev_p$ on each of the $x_i$. Linearity gives that the homomorphisms agree on scalars too, so they agree on any polynomial.
Thus $\psi = ev_p$.
Finally, let $f$ be in the ideal defining $X$. $\psi(f) = \psi(0) = 0$. Thus $p \in X$.