Schoenberg (Ann. of Math. 39 (1938), 811-841) observed that if $f(t)$ is a completely monotonic function, then the radial kernel $K(x,y)=f(\|x-y\|^2)$ is positive definite on any Hilbert space. Since $$ f(t):=\frac{1}{1+\frac{t}{\sigma^2}} $$ is completely monotonic, the result follows.

Added. Schoenberg's proof relies on the Hausdorff-Bernstein-Widder theorem and the fact that the Gaussian kernel $\exp(-\|x-y\|^2)$ is positive definite. Using these two facts, the proof is immediate.

Schoenberg (Ann. of Math. 39 (1938), 811-841) observed that if $f(t)$ is a completely monotonic function, then the radial kernel $K(x,y)=f(\|x-y\|^2)$ is positive definite on any Hilbert space. Since $$ f(t):=\frac{1}{1+\frac{t}{\sigma^2}} $$ is completely monotonic, the result follows.

Added. Schoenberg's proof relies on the Hausdorff-Bernstein-Widder theorem and the fact that the Gaussian kernel $\exp(-\|x-y\|^2)$ is positive definite. Using these two facts, the proof is immediate.

For a modern account, see Theorem 7.13 in Wendland: Scattered Data Approximation (Cambridge University Press, 2005).

Schoenberg (Ann. of Math. 39 (1938), 811-841) observed that if $f(t)$ is a completely monotonic function, then the radial kernel $K(x,y)=f(\|x-y\|^2)$ is positive definite on any Hilbert space. Since $$ f(t):=\frac{1}{1+\frac{t}{\sigma^2}} $$ is completely monotonic, the result follows.

Schoenberg (Ann. of Math. 39 (1938), 811-841) observed that if $f(t)$ is a completely monotonic function, then the radial kernel $K(x,y)=f(\|x-y\|^2)$ is positive definite on any Hilbert space. Since $$ f(t):=\frac{1}{1+\frac{t}{\sigma^2}} $$ is completely monotonic, the result follows.

Added. Schoenberg's proof relies on the Hausdorff-Bernstein-Widder theorem and the fact that the Gaussian kernel $\exp(-\|x-y\|^2)$ is positive definite. Using these two facts, the proof is immediate.