In theory, the kernel of two vectors is equal to the dot-product of them in feature space, i.e.

k(x1,x2)=, but practically this seems tricky!

Suppose that we trained a kernel mapping such as Kernel PCA (KPCA) on some data. We can obtain image of any vector x in feature space as phi(x) using the trained mapping. but when I evaluate the above equation on real data, the equality does not hold true!

Am I missing some facts?

(I think this is related to empirical kernel map)