Proposition 3. Let $\rho: G \to \operatorname{GL}(V)$ be a linear representation of $G$, and let $\chi$ be its character. Let $\chi^2_\sigma$ be the character of the symmetric square $\operatorname{Sym}^2(V)$ of $V$, and let $\chi_\alpha^2$ be that of $\operatorname{Alt}^2(V)$. For each $s \in G$, we have \begin{align} \chi_\sigma^2 &= \frac{1}{2} (\chi(s)^2 + \chi(s^2)) \\ \chi_\alpha^2 &= \frac{1}{2} (\chi(s)^2 - \chi(s^2)) \\ \end{align} and $\chi_\sigma^2 + \chi_\alpha^2 = \chi^2$.
Serre starts with his proof taking a basis consisting of eigenvectors of $\rho_s$ and justified it with the observation that $\rho_s$ can be represented by a unitary matrix and refers to subsection 1.3. Unitary matrices are diagonalizable by the Spectral Theorem, as unitary matrices are normal too, so I am okay with this part.
In mentioned subsection, we assumed that the vector $V$ was endowed with a scalar product. However, this is not mentioned in the hypothesis of Proposition 3.
Could someone explain me if the argument is wrong resp. incomplete? Thank you!