Exercise 12.5: Kernels and Cauchy–Schwarz
(a)
If $k$ is positive definite, then \begin{equation} \det \begin{bmatrix} k(x, x) & k(x, y) \newline k(y, x) & k(y, y) \end{bmatrix} = k(x, x) k(y, y) - k^2(x, y) \ge 0 \implies k(x, y) \le \sqrt{k(x, x) k(y, y)}. \end{equation}
(b)
Set $k(f, g) = \lra{f, g}$. We verify that $k$ is positive definite: \begin{equation} \sum_{i=1}^n \sum_{j=1}^n \alpha_i \alpha_j \lra{f_i, f_i} = \lra{ \sum_{i=1}^n \alpha_i f_i, \sum_{i=1}^n \alpha_i f_i } \ge 0. \end{equation} The result then follows from (a).