id: 05710100
dt: j
an: 05710100
au: Edalat, Abbas
ti: A differential operator and weak topology for Lipschitz maps.
so: Topology Appl. 157, No. 9, 1629-1650 (2010).
py: 2010
pu: Elsevier Science B.V. (North-Holland), Amsterdam
la: EN
cc: 
ut: domain theory; Clarke gradient; weakest topology; second order functionals;
    Hausdorff metric; fundamental theorem of calculus
ci: 
li: doi:10.1016/j.topol.2010.03.003
ab: 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.
rv: Teodora-Liliana Rădulescu (Craiova)