Paper: Weak Topology and a Differentiable Operator for Lipschitz Maps (at LICS 2008)

Authors: Abbas Edalat

Abstract

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 characterisation of the basic open subsets of the L-topology in terms of ties or primitive maps of functions. We use this to verify that the L-topology is strictly coarser than the well-known 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.

BibTeX

  @InProceedings{Edalat-WeakTopologyandaDif,
    author = 	 {Abbas Edalat},
    title = 	 {Weak Topology and a Differentiable Operator for Lipschitz Maps},
    booktitle =  {Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science (LICS 2008)},
    year =	 {2008},
    month =	 {June}, 
    pages =      {364--375},
    location =   {Pittsburgh, PA, USA}, 
    publisher =	 {IEEE Computer Society Press}
  }