theorem:kernel-function-iff-embedding

Kernel Is Embedding Theorem. A function $k : \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ is a kernel function if and only if there exists a Hilbert space and a map $\Phi : \mathcal{X} \to \mathcal{H}$ such that $k(x,y) = \langle\Phi(x),\Phi(y)\rangle$.