Computational complexity of classifying with an already-trained SVM
If you're classifying using SVMs, then the underlying metric space is always a Hilbert space. If your classifier is linear, then the running time of the classification is linear in the dimensionality of the data (or the number of features).

If the classifier (in general) involves some kernel, then the classifier is expressed in terms of the number of support vectors, and the classification is linear in the number of such vectors. Processing each vector takes time proportional to the kernel computation time (which could be constant, or linear in the data dimension, or something else)