Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 4 / a / Solution

A real positive-semidefinite kernel is a symmetric function $k:X\times X\to \mathbb{R}$ such that every finite Gram matrix $(k(x_i,x_j))$ is positive semidefinite. The Moore-Aronszajn theorem says that there are a Hilbert space $H$ and a feature map $\varphi :X\to H$ such thatEquivalently, $H$ can be chosen as the unique Reproducing-kernel Hilbert space with reproducing kernel $k$.