See problem 3.26 in Etingof's "Introduction to representation theory". If you have troubles with understanding the hint, feel free to ask me. (The first sentence uses the fact that if a vector space over an infinite field is the union of finitely many subspaces, then one of these subspaces is the whole vector space.)

See problem 3.26 in Etingof's "Introduction to representation theory". If you have troubles with understanding the hint, feel free to ask me. (The first sentence uses the fact that if a vector space over an infinite field is the union of finitely many subspaces, then one of these subspaces is the whole vector space. The surjectivity of the map $SV\to F\left(G,\mathbb C\right)$ is because a polynomial can take any arbitrary finite set of values at some given distinct points. In order to conclude from this, note that this map $SV\to F\left(G,\mathbb C\right)$ is a homomorphism of representations of $G$.)

This proof works over any algebraically closed field of characteristic $0$. This can't quite be said about the proof in Fulton-Harris, if I remember it right.