Let \(\mathscr{G}_{0}\) be the pre-RKHS \(\mathscr{G}_{0}\) constructed in the proof of Theorem 6.2. Thus, \(g \in \mathscr{G}_{0}\) is of the form \(g=\sum_{i=1}^{n} \alpha_{i} \kappa_{x_{i}}\) and

\[ \left\langle g, \kappa_{\boldsymbol{x}}\rightangle_{\mathscr{G}_{0}}=\sum_{i=1}^{n} \alpha_{i}\left\langle\kappa_{\boldsymbol{x}_{i}}, \kappa_{\boldsymbol{x}}\rightangle_{\mathscr{G}_{0}}=\sum_{i=1}^{n} \alpha_{i} \kappa\left(\boldsymbol{x}_{i}, \boldsymbol{x}\right)=g(\boldsymbol{x}) \]

Therefore, we may write the evaluation functional of \(g \in \mathscr{G}_{0}\) at \(\mathrm{x}\) as \(\delta_{x} g:=\left\langle g, \kappa_{x}\rightangle_{\mathscr{G}_{0}}\), Show that \(\delta_{\boldsymbol{x}}\) is bounded on \(\mathscr{G}_{0}\) for every \(\boldsymbol{x}\); that is, \(\left|\delta_{\boldsymbol{x}} f\right|<\gamma\|f\|_{\mathscr{G}_{0}}\), for some \(\gamma<\infty\).