For a positive definite kernel, there exists a Reproducing Kernel Hilbert Space (RKHS) whose reproducing kernel is the kernel function.
My question is, what characteristic of the kernel function decides the inner product of the RKHS. For example, for the second order polynomial kernel $k(x, x') = (x^Tx')^2$ where $x, x' \in \mathbb{R}^2$, its feature map is $\phi_x = k(\cdot,x) = [x_1, \sqrt2x_1x_2, x_2]$ and $\langle\cdot,\cdot\rangle$ is computed in the same way as Euclidean space.
However, for Gaussian kernel the feature map is a mapping to infinite dimensional space, and the inner product will be computed as an integral.
For those well-known kernels we know the feature map, hence we can think about the inner product of the corresponding RKHS, but when we came up with a random positive definite kernel, what characteristics of the kernel function decide the inner product and the feature map?