Mercer Theorem

$$\mathbf{K}=\left(\begin{array}{ccc}k(x_1,x_1)& \dots & k(x_1,x_N)\\ ⋮& \ddots & ⋮\\ k(x_N,x_1)& \dots & k(x_N,x_N)\end{array}\right)$$

• $k(x_n,x_m)$ is a kernel function iff $\forall \text{finite }N\text{ and }{x}_1,\dots ,x_N$, this matrix is a positive semidefinite matrix