A differential operator and weak topology for Lipschitz maps (Q972498)

One of the first main results of this paper asserts that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces. This is usually referred to as the \(L\)-topology and is the weakest topology with respect to which the \(L\)-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the \(L\)-derivative and the Clarke gradient coincide, the author provides a simple characterization of the basic open subsets of the \(L\)-topology. The present paper also develops a fundamental theorem of calculus of second order in finite dimensions. A key ingredient in the proof is that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the \(L\)-derivative. In the last part of this paper it is shown that in dimension one the \(L\)-derivative operator is a computable functional.