Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 3 / a / Solution

A symmetric function $k:{\mathcal{X}}^2\to \mathbb{R}$ is a positive-definite kernel when ${\sum }_{i,j}{a}_i{a}_{j}k(x_i,x_j)\ge 0$ for every finite set of points and real coefficients. A Reproducing-kernel Hilbert space $\mathcal{H}$ is a Hilbert space of functions whose evaluation maps are continuous. Its reproducing kernel satisfies $k(x,\cdot )\in \mathcal{H}$ and $f(x)=⟨f,k(x,\cdot ){⟩}_{\mathcal{H}}$.