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This article studies connections between group actions and theircorresponding 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 itsfixed subspace: $$ L^G(X) = \{f\in L(X)\colon f(a\cdot x) = f(x) \textrm{ forall }a\in G, x\in X\}. $$ In particular, we show that $L^G(X)$ is an invariantof the action of $G$ on $X$. In the case when the action is finite, we computethe dimension of $L^G(X)$ in terms of fixed points of $X$ and prove severalprominent results for $L^G(X)$, including Bessel's inequality and Frobeniusreciprocity.

Frobenius reciprocity on the space of functions invariant under a group action