The standard inner product on $\mathrm{c}\mathrm{f}(G)$

For two class functions $\phi ,\psi \in \mathrm{c}\mathrm{f}(G)$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$), define

$⟨\phi ,\psi ⟩:=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{g\in G}\phi (g)\overline{\psi (g)},$

the finite sum being that of (A finite sum in a commutative monoid indexed by an arbitrary finite set) for the additive commutative monoid of $\mathbb{C}$. This is the standard Hermitian form on the complex vector space of class functions.

This assignment is an inner product in the exact sense of the published definition (Real and complex inner product spaces, with the inner product linear in the first argument), with the inner product linear in the first argument. Linearity in the first slot is immediate from the pointwise vector-space structure of $\mathrm{c}\mathrm{f}(G)$ and linearity of the finite sum. The conjugate-symmetry clause holds because conjugation (Real and imaginary parts, complex conjugation, and modulus) is an involution: $⟨\psi ,\phi ⟩=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g\psi (g)\overline{\phi (g)}=\overline{\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g\phi (g)\stackrel{‾}{\psi (g)}}=\overline{⟨\phi ,\psi ⟩}$. For definiteness, $⟨\phi ,\phi ⟩=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_g\mathrm{\mid }\phi (g){\mathrm{\mid }}^2$ is a sum of nonnegative real numbers, hence nonnegative real; it is $0$ exactly when every term $\mathrm{\mid }\phi (g){\mathrm{\mid }}^2$ is $0$, and since $\mathrm{\mid }z\mathrm{\mid }=0$ holds only for $z=0$ in $\mathbb{C}$, that says exactly $\phi =0$ as a function.