For a kernel function, we have two conditions one is that it should be symmetric which is easy to understand intuitively because dot products are symmetric as well and our kernel should also follow this. The other condition is given below

There exists a map $ φ:R^d→H$ called kernel feature map into some high dimensional feature space H such that $ ∀x,x′$ in $ R^d:k(x,x′)=<φ(x),φ(x′)>$

I understand that this means that there should exist a feature map that will project the data from low dimension to any high dimension D and kernel function will take the dot product in that space.

For example, the Euclidean distance is given as

$ d(x,y)=∑_i(x_i−y_i)^2=<x,x>+<y,y>−2<x,y>$

If I look this in terms of the second condition how do we know that doesn’t exist any feature map for euclidean distance? What exactly are we looking in feature maps mathematically?