Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$

Let $G$ be a finite group. A function $f:G\to \mathbb{C}$ is a class function when it is constant on every conjugacy class (The conjugacy class ${\mathrm{Cl}}_G(x)$ and centralizer $C_G(x)$ of an element):

$f(gx{g}^{-1})=f(x)\text{for all }g,x\in G\mathrm{.}$

The set of all class functions is written

$\mathrm{c}\mathrm{f}(G):=\{f:G\to \mathbb{C}:f(gx{g}^{-1})=f(x)\text{ for all }g,x\in G\}\mathrm{.}$

With pointwise addition $(f+h)(g):=f(g)+h(g)$ and pointwise scalar multiplication $(\lambda f)(g):=\lambda f(g)$, the set $\mathrm{c}\mathrm{f}(G)$ is a complex vector space (Vector space over a field): the axioms (V1)–(V5) are inherited from the field $\mathbb{C}$ at every argument $g\in G$. Because $G$ is finite, the evaluation functions $f\mapsto f(x)$ are complex-linear, and a class function is determined by its values on one representative of each conjugacy class.