On the fixed space induced by a group action

This article studies connections between group actions and their corresponding vector spaces. Given an action of a group $G$ on a non-empty set $X$, we examine the space $L(X)$ of scalar-valued functions on $X$ and its fixed subspace: $L^G(X)=\{f\in L(X):f(a\cdot x)=f(x)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}a\in G,x\in X\}$. In particular, we show that $L^G(X)$ is an invariant of the action of $G$ on $X$. In the case when the action is finite, we compute the dimension of $L^G(X)$ in terms of fixed points of $X$ and prove several prominent results for $L^G(X)$, including Bessel's inequality and Frobenius reciprocity.