My attempt:
Let $x,y\in\mathbb{R}^d$. We already know the Fourier transform of a Gaussian function is a Gaussian function.If substituting $x-y$ for the variable after Fourier transform, we have $$ \exp\left(-\frac{\|x-y\|^2}{2\sigma^2}\right)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\exp\left(\frac{i \xi^\mathrm{T} x}{\sigma}\right)\exp\left(-\frac{i \xi^\mathrm{T} y}{\sigma}\right)\exp\left(-\frac{\|\xi\|^2}{2}\right)\,\mathrm{d}\xi $$ which means $\exp[-\|x-y\|^2/(2\sigma^2)]$ is an inner product with weight $\exp(-\|\xi\|^2/2)$.
Secondly, $\{\exp[(i \xi^\mathrm{T} x)/\sigma]\}_{\xi\in\mathbb{R}^d}$ are basis of Fourier transform, which implies linear independence, so they can be seen as the basis of the implicit feature space (remember they form the inner product). They are uncountable because the subscript $\xi$ is continuous.
The problem is that from Mercer's theorem, a symmetric positive semidefinite kernel should admit countably many basis vectors. So there should not be uncountably many basis vectors.
What is wrong in my attempt of deriving the basis of the implicit feature space of Gaussian kernel?
