Compact families of piecewise constant functions in Lp(0,T;B)Lp(0,T;B)

A strong compactness result in the spirit of the Lions–Aubin–Simon lemma is proven for piecewise constant functions in time (uτ)(uτ) with values in a Banach space. The main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts uτ−uτ(⋅−τ)uτ−uτ(⋅−τ) instead of all time shifts uτ−uτ(⋅−h)uτ−uτ(⋅−h) for h>0h>0, as required in Simon’s compactness theorem. This simplifies significantly the application of the Rothe method in the existence analysis of parabolic problems.