Exercise 12.5: Kernels and Cauchy–Schwarz

chapter 12

(a)

If $k$ is positive definite, then $$\begin{array}{}\text{(1)}& \det \left[\begin{array}{cc}k(x,x)& k(x,y)\\ k(y,x)& k(y,y)\end{array}\right]=k(x,x)k(y,y)-k^2(x,y)\ge 0⟹k(x,y)\le \sqrt{k(x,x)k(y,y)}.\end{array}$$

(b)

Set $k(f,g)=⟨f,g⟩$. We verify that $k$ is positive definite: $$\begin{array}{}\text{(2)}& \sum _{i=1}^n\sum _{j=1}^n{\alpha }_i{\alpha }_j⟨f_i,f_i⟩=⟨\sum _{i=1}^n{\alpha }_i{f}_i,\sum _{i=1}^n{\alpha }_i{f}_i⟩\ge 0.\end{array}$$ The result then follows from (a).