Definition 1

A function $f\in \mathscr{C}(X\times X)$ is said to be a universally positive kernel on X if it satisfies the equivalent conditions of Theorem 1.

If $f$ is a universally positive kernel on X, a pair $(E, g)$ satisfying condition (iv) of loc. cit. is called a Hilbert realization of $f$; if condition (iii) is satisfied, it is said to be a cyclic Hilbert realization.

We denote by Noy$_+(X)$ the set of universally positive kernels on X.

Let $X'$ be a separated topological space and $h: X\rightarrow X'$ a continuous mapping. The mapping $f\mapsto f\circ (h, h)$ from $\mathscr{C}(X'\times X')$ into $\mathscr{C}(X\times X)$ induces, by passing to the subspaces, a mapping Noy$_+(X')\rightarrow$ Noy$_+(X)$.