Existence of variational source conditions for nonlinear inverse problems in Banach spaces

Variational source conditions proved to be useful for deriving convergence rates for Tikhonov's regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.