In nonparametric classification and regression problems, regularized kernel methods, in particular support vector machines, attract much attention in theoretical and in applied statistics. In an abstract sense, regularized kernel methods (simply called SVMs here) can be seen as regularized M-estimators for a parameter in a (typically infinite dimensional) reproducing kernel Hilbert space. For smooth loss functions LL, it is shown that the difference between the estimator, i.e. the empirical SVM View the MathML sourcefL,Dn,λDn, and the theoretical SVM fL,P,λ0fL,P,λ0 is asymptotically normal with rate View the MathML sourcen. That is, View the MathML sourcen(fL,Dn,λDn−fL,P,λ0) converges weakly to a Gaussian process in the reproducing kernel Hilbert space. As common in real applications, the choice of the regularization parameter View the MathML sourceDn in View the MathML sourcefL,Dn,λDn may depend on the data. The proof is done by an application of the functional delta-method and by showing that the SVM-functional P↦fL,P,λP↦fL,P,λ is suitably Hadamard-differentiable.