Let <math>X</math> denote a random variable with codomain <math>\Omega</math> and distribution <math>P(Xdx).</math> Given a kernel <math>k</math> on <math>\Omega \times \Omega,</math> the [[Reproducing kernel Hilbert space#Moore–Aronszajn theorem|Moore–Aronszajn theorem]] asserts the existence of a RKHS <math>\mathcal{H}</math> (a [[Hilbert space]] of functions <math>f: \Omega \to \R</math> equipped with inner products <math>\langle \cdot, \cdot \rangle_\mathcal{H}</math> and norms <math>\| \cdot \|_\mathcal{H}</math>) in which the element <math>k(x,\cdot)</math> satisfies the reproducing property
:<math>\forall f \in \mathcal{H}, \forall x \in \Omega \qquad \langle f, k(x,\cdot) \rangle_\mathcal{H} = f(x).</math>
