We study direct estimates for adaptive time-stepping finite element methods for time-dependent partial differential equations. Our results generalize previous findings from “On approximation classes for adaptive time-stepping finite element methods” by Actis et al. (2023), where the approximation error was only measured in L 2 ( [ 0 , T ] , L 2 ( Ω ) ) . In particular, we now also cover the error norms L ∞ ( [ 0 , T ] , L 2 ( Ω ) ) and L 2 ( [ 0 , T ] , H 1 ( Ω ) ) which are more natural in this context.