Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 1 / Solution

A positive-semidefinite kernel on a nonempty set $\mathcal{X}$ is a symmetric function $k:\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ such that, for every $n\ge 1$, every $x_1,\dots ,x_n\in \mathcal{X}$, and every $c_1,\dots ,c_n\in \mathbb{R}$,Equivalently, every finite kernel matrix $(k(x_r,x_s){)}_{r,s}$ is a positive semidefinite matrix.