On ℒ2, the set of Lipschitz continuous operators is a set of first category in the set of uniformly continuous operators

In this paper, we prove that the set of Lipschitz continuous nonlinear operators defined from ℒ 2 into ℒ 2 is dense on the set of uniformly continuous operators and is furthermore a set of first category in the same set. It is the extension of a well-known result in the case of continuous real-valued functions to the case of the (nonlinear) operators. This result is important for the control community due to the interest of Lipschitz continuous operators and the associated incremental approach for the development of nonlinear control.