Exercise 5, § 3

Let X be a separated topological space. A continuous function $f: X\times X\rightarrow \mathbf{R}$ is said to be conditionally of negative type if it satisfies the conditions

(i) For every $x\in X$, one has $f(x, x) = 0$;

(ii) For all $x$ and $y$ in X, one has $f(x, y) =f(y, x)$;

(iii) For every integer $n\in \mathbf{N}$, every family $(x_i)_{0\leqslant i\leqslant n}$ of elements of X and every family $(t_i)_{0\leqslant i\leqslant n}$ of real numbers such that

one has

Denote by Noy$_-(X)$ the set of functions conditionally of negative type on $X\times X$.

a) Let E be a real Hilbert space and $\varphi : X\rightarrow E$ a continuous mapping. The function $f(x, y) =\|\varphi (x)-\varphi (y)\|^2$ is conditionally of negative type. b) Let $f\in$ Noy$_-(X)$ and $x_0\in X$. There exist a real Hilbert space E and a continuous mapping $\varphi : X\rightarrow E$ such that

a) For every $(x, y)\in X\times X$, one has $f(x, y) =\|\varphi (x)-\varphi (y)\|^2$;

b) The family $(\varphi (x)-\varphi (x_0))_{x\in X}$ is total in E.

One says that $(E, \varphi )$ is a cyclic Hilbertian representation of $f$.

c) Let $f\in$ Noy$_-(X)$ and $x_0\in X$. Let $(E, \varphi )$ and $(E', \varphi ')$ be cyclic Hilbertian representations of $f$. There exists a unique affine isometry $u: E\rightarrow E'$ such that $\varphi '=u\circ \varphi$.

d) The set Noy$_-(X)$ is a convex cone in $\mathscr{C}(X\times X;\mathbf{R})$. For every universally positive kernel $g\in$ Noy$_+(X)$ such that $g$ is real-valued, the mapping $f(x, y) =g(x, x)-g(x, y)$ belongs to Noy$_-(X)$.

e) Let $\mathfrak{F}$ be a filter on Noy$_-(X)$ which converges simply to $f: X\times X\rightarrow \mathbf{R}$. If $f$ is continuous, then $f\in$ Noy$_-(X)$.

f ) Let $f: X\times X\rightarrow \mathbf{R}$ be a continuous function satisfying conditions (i) and (ii) above. Let $x_0\in X$. Let $g: X\times X\rightarrow \mathbf{R}$ be such that $g(x, y) =f(x, x_0) +$ $f(y, x_0)-f(x, y)$. Then $f\in$ Noy$_-(X)$ if and only if $g\in$ Noy$_+(X)$.

g) Suppose X nonempty. Let $f: X\times X\rightarrow \mathbf{R}$ be a continuous function satisfying conditions (i) and (ii) above. Then $f\in$ Noy$_-(X)$ if and only if one has exp($-tf$)$\in$ Noy$_+(X)$ for all $t\in \mathbf{R}_+$ ("Schoenberg's theorem"). (To prove that exp($-f$) is of positive type when $f\in$ Noy$_-$(X), fix $x_0\in X$; prove that

$g(x, y) =$ exp($f(x, x_0) +f(y, x_0)-f(x, y)$)

and

$h(x, y) =$ exp($-f(x, x_0)$) exp($-f(y, x_0)$)

belong to Noy$_+(X).)$

If G is a topological group, a function $\varphi \in \mathscr{C}(G)$ is said to be of negative type if the function $f$ on $G\times G$ defined by $f(g, h) =\varphi (h^{-1}g)$ is a conditionally negative type kernel.