How general are general source conditions?

Error analysis of regularization methods in Hilbert spaces is based on smoothness assumptions in terms of source conditions. In the traditional setup, i.e. when smoothness is in a power scale, we see that not all elements in the underlying Hilbert space possess some smoothness with this scale. Our main result asserts that this can be overcome when turning to general source conditions defined in terms of index functions. We conclude with some consequences.