Finally, what happens if one uses a kernel which does not satisfy Mercer’s condition? In general, there may exist data such that the Hessian is indefinite, and for which the quadratic programming problem will have no solution (the dual objective function can become arbitrarily large). However, even for kernels that do not satisfy Mercer’s condition, one might still find that a given training set results in a positive semidefinite Hessian, in which case the training will converge perfectly well. In this case, however, the geometrical interpretation described above is lacking."
Burgess (1998)