A differential operator and weak topology for Lipschitz maps

作者：Edalat Abbas^*

来源：Topology and Its Applications, 2010, 157(9): 1629-1650.

DOI：10.1016/j.topol.2010.03.003

• Domain theory
• Clarke gradient
• Weakest topology
• Second order functionals
• Hausdorff metric
• Fundamental Theorem of Calculus

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It 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, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing 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. We finally show that in dimension one the L-derivative operator is a computable functional.

• 出版日期2010-6-15