Length and Mean Value Theorem in Norm are the flip sides of the same coin

In this paper, we reveal the links between two approaches that both use linearizations of a nonlinear system in order to investigate its properties: the mean value theorem in norm and the length approach in the input-output context. We first prove that the length approach, used as the basis for the contraction approach, can also be applied to characterize the properties of operators defined between (infinite dimensional) functional spaces. We then prove that, as in the finite dimension, these two approaches are in fact intertwined and in a way form the flip sides of the same coin.