By a valid kernel, we mean the Gram matrix induced by it is positive semi-definite (and symmetric).
Let $K(x,y)=\exp(-\lambda \|x-y \|)$, where $\lambda >0.$ We say $K$ is a Laplacian kernel.
I know the exponential of a valid kernel is still a valid kernel. But unfortunately, $K_1(x,y)=\|x-y \|$ is not a valid kernel and my strategy is not working.
I have googled a lot and found a solution involving Fourier transform and reproducing kernel Hilbert spaces(abbr. RKHS). But I am new to this area and not really familiar with RKHS. I wonder if there is an elementary way to prove this. Many thanks.