  In recent years, a series of convergence rates conditions for regularization
methods has been developed. Mainly, the motivations for developing novel
conditions came from the desire to carry over convergence rates results from
the Hilbert space setting to generalized Tikhonov regularization in Banach
spaces. For instance, variational source conditions have been developed and
they were expected to be equivalent to standard source conditions for linear
inverse problems in a Hilbert space setting. We show that this expectation does
not hold. However, in the standard Hilbert space setting these novel conditions
are optimal, which we prove by using some deep results from Neubauer, and
generalize existing convergence rates results. The key tool in our analysis is
a novel source condition, which we put into relation to the existing source
conditions from the literature. As a positive by-product, convergence rates
results can be proven without spectral theory, which is the standard technique
for proving convergence rates for linear inverse problems in Hilbert spaces.
