A differential operator and weak topology for Lipschitz maps
type: Item
instance of: scholarly article
title: A differential operator and weak topology for Lipschitz maps
zbMATH Open document ID: 1195.26034
author: Abbas Edalat
published in: Topology and its Applications
publication date: 2010-05-19T00:00:00Z
full work available at URL: http://hdl.handle.net/10044/1/14154
review text: 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.
reviewed by: Teodora-Liliana Rădulescu
Mathematics Subject Classification ID: 26A36
Mathematics Subject Classification ID: 03D80
Mathematics Subject Classification ID: 26B12
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 5710100
zbMATH Keywords: domain theory
zbMATH Keywords: Clarke gradient
zbMATH Keywords: weakest topology
zbMATH Keywords: second order functionals
zbMATH Keywords: Hausdorff metric
zbMATH Keywords: fundamental theorem of calculus
MaRDI profile type: Publication
OpenAlex ID: W2062308906
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DOI: 10.1016/J.TOPOL.2010.03.003
description: scientific article; zbMATH DE number 5710100