Exercises 4-6 show that \(\mathscr{G}\) defined in the proof of Theorem 6.2 is an inner product space. It remains to prove that \(\mathscr{G}\) is an RKHS. This requires us to prove that the inner product space \(\mathscr{G}\) is complete (and thus Hilbert), and that its evaluation functionals are bounded and hence continuous (see Theorem A.16). This is done in a number of steps.

(a) Show that \(\mathscr{G}_{0}\) is dense in \(\mathscr{G}\) in the sense that every \(f \in \mathscr{G}\) is a limit point (with respect to the norm on \(\mathscr{G})\) of a Cauchy sequence \(\left(f_{n}\right)\) in \(\mathscr{G}_{0}\).

(b) Show that every evaluation functional \(\delta_{x}\) on \(\mathscr{G}\) is continuous at the 0 function. That is,

\[ \begin{equation*} \forall \varepsilon>0: \exists \delta>0: \forall f \in \mathscr{G}:\|f\|_{\mathscr{G}}<\delta \Rightarrow|f(\mathbf{x})|<\varepsilon \tag{6.40} \end{equation*} \]

Continuity of \(\delta_{\mathrm{x}}\) at all functions \(g \in \mathscr{G}\) then follows automatically from linearity.

(c) Show that \(\mathscr{G}\) is complete; that is, every Cauchy sequence \(\left(f_{n}\right) \in \mathscr{G}\) converges in the norm \(\|\cdot\|_{\mathscr{G}}\).