Suppose an (mathrm{RKHS} mathscr{G}) of functions from (mathscr{X} ightarrow mathbb{R}) (with kernel (kappa) ) is invariant

Suppose an \(\mathrm{RKHS} \mathscr{G}\) of functions from \(\mathscr{X} \rightarrow \mathbb{R}\) (with kernel \(\kappa\) ) is invariant under a group \(\mathscr{T}\) of transformations \(T: \mathscr{X} \rightarrow \mathscr{X}\); that is, for all \(f, g \in \mathscr{G}\) and \(T \in \mathscr{T}\), we have (i) \(f \circ T \in \mathscr{G}\) and (ii) \(\langle f \circ T, g \circ Tangle_{\mathscr{G}}=\langle f, gangle_{\mathscr{\varphi}}\). Show that \(\kappa\left(T \boldsymbol{x}, \boldsymbol{T} \boldsymbol{x}^{\prime}\right)=\kappa\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right)\) for all \(\boldsymbol{x}, \boldsymbol{x}^{\prime} \in \mathscr{X}\) and \(T \in \mathscr{T}\).