Isomorphism from $\mathbb{C}[G]$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.

What I want to ask is the proof of the Proposition 10. in "Linear Representations of Finite Groups" by Jean-Pierre Serre.

Let $\rho_i : G \rightarrow GL(W_i)$ be the distinct irreducible representations of $G$.$(1 \le i \le h)$ We can extend $\rho_i$ to $\widetilde{\rho_i} : \mathbb{C}[G] \rightarrow \text{End}(W_i)$ by $\widetilde{\rho_i}(\sum_{g \in G} a_g g) = a_g \sum_{g \in G} \rho_i(g)$.

Then $\widetilde{\rho} = (\widetilde{\rho_i}) : \mathbb{C}[G] \rightarrow \prod_{i=1}^{h} \text{End}(W_i) \cong \prod_{i=1}^h M_{n_i}(\mathbb{C})$ is a homomorphism.

Then Proposition 10 states that it is an isomorphism. Since dimensions of domain and codomain are same, it suffices to show that $\widetilde{\rho}$ is surjective.

In the proof of the book,

If $\widetilde{\rho}$ is not surjective, then there exists a nonzero linear form on $\prod M_{n_i}(\mathbb{C})$ vanishing on the image of $\widetilde{\rho}$. This gives a nontrivial relation on the coefficients of the representations $\rho_i$ which is impossible because of the orthogonality formulas of 2.2.

But I can not follow the above three lines in the proof. I will appreciate it if you give an explanation for the proof.

Thank you.

Given any proper subspace $U\subset V$ and any $v\in V$, there exists a linear function $\phi:V\to\Bbb C$ which satisfies $\phi(u)=0$ for all $u\in U$ and $\phi(v)\ne0$. To see this, pick a basis for $U$, adjoin $v$ to it and extend to a basis for all of $V$, then consider the coordinate projection associated to $v$.

Another fact to know: Given a coordinate vector space $V$, any linear map $V\to\Bbb C$ is a linear combination of coordinate projection maps.

Consider $\Bbb C[G]\to\prod M_{n_i}(\Bbb C)$. Suppose the image is proper, so there exists a nontrivial linear functional on the latter space which vanishes on the image. As the functional is linear, it must be some linear combination of the coordinates of all of the matrices. Therefore,

$$\sum_{i,j,k} a_{ijk}(\rho_i(g))_{jk}=0 \tag{$\circ$}$$

for all $g\in G$, where $(\rho_i(g))_{jk}$ is the $j,k$ matrix coefficient of $\rho_i(g)$ and $a_{ijk}$ are the coefficients present in the linear functional. But if $(\circ)$ is true for all $g$, then it is true when $(\circ)$ is understood as a linear combination of functions $G\to\Bbb C$, or in other words the matrix coefficients are linearly dependent functions, since they satisfy the dependence relation $(\circ)$. But this is impossible.

If you're interested in a different proof, I've written one here, which I think is slick, and doesn't make reference to matrix coefficients. (It is an "abstract nonsense" proof which leverages representability of the fiber functor, to couch the elementary proof in a deeper language.) It uses Maschke's theorem that complex representations of finite groups are semisimple, and Schur's lemma to compute dimensions of intertwiners. It also explains Schur's orthogonality relations for characters.