Let $x ∈ S^2$ be an arbitrary point on the unit sphere. In addition, let $L_R$ be the rotation operator taking a function $f$ and generating a rotated function $L_Rf$ by $[L_Rf](x) = f(R^{−1}x)$, where $R ∈SO(3)$.
I want to show that $L_{R^{−1}}$ is adjoint to $L_R$.
I have tried the following:
$[L_{R^{-1}}L_Rf](x)=[L_{R^{-1}}f(R^{-1}x)](x) = f(RR^{-1}x) = f(x)$
And therefore $L_{R^{−1}}$ is adjoint to $L_R$
Am I missing something? Is it realy straightforward?