 Solution

A symmetric function $k:\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ is a positive-semidefinite kernel when, for every $m$, every $x_1,\dots ,x_m\in \mathcal{X}$, and every $c\in {\mathbb{R}}^m$,Equivalently, every associated kernel matrix is a positive semidefinite matrix.

A Reproducing-kernel Hilbert space $\mathcal{H}$ on $\mathcal{X}$ is a Hilbert space of real-valued functions such that every point-evaluation map $f\mapsto f(x)$ is continuous. The Riesz representation theorem then gives a function $k(x,\cdot )\in \mathcal{H}$ satisfying the reproducing propertyIts reproducing kernel is $k(x,y)=⟨k(x,\cdot ),k(y,\cdot ){⟩}_{\mathcal{H}}$.