Positive-definite kernel

Summary

A positive-definite kernel on a set $X$ is a function $K \colon X \times X \to \C$ with the following property. Every matrix $\big(K(x_i, x_j)\big)_{i,j}$ built from finitely many points of $X$ is positive semidefinite. It generalises positive semidefinite matrices from finite index sets to arbitrary sets. Each positive-definite kernel is the reproducing kernel of a unique reproducing kernel Hilbert space, and conversely. On a group, the kernels $K(g, h) = \varphi(g^{-1}h)$ that are positive-definite are those given by positive-definite functions $\varphi$.