  This article studies connections between group actions and their
corresponding vector spaces. Given an action of a group $G$ on a nonempty set
$X$, we examine the space $L(X)$ of scalar-valued functions on $X$ and its
fixed subspace: $$ L^G(X) = \{f\in L(X)\colon f(a\cdot x) = f(x) \textrm{ for
all }a\in G, x\in X\}. $$ 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.
