## Weihrauch, Klaus and Jafarikhah, Tahereh - Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$

lmcs:1117 - Logical Methods in Computer Science, September 2, 2014, Volume 10, Issue 3
Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$

Authors: Weihrauch, Klaus and Jafarikhah, Tahereh

By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation from the unit interval into the reals, or by signed measures on the Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition into non-negative or non-decreasing objects. Using the representation approach to computable analysis, a computable version of the Riesz representation theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we extend this result. We study the computable relation between three Banach spaces, the space of linear continuous functionals with operator norm, the space of (normalized) functions of bounded variation with total variation norm, and the space of bounded signed Borel measures with variation norm. We introduce natural representations for defining computability. We prove that the canonical linear bijections between these spaces and their inverses are computable. We also prove that Jordan decomposition is computable on each of these spaces.

Source : oai:arXiv.org:1407.3679
DOI : 10.2168/LMCS-10(3:13)2014
Volume: Volume 10, Issue 3
Published on: September 2, 2014
Submitted on: October 7, 2013
Keywords: Computer Science - Logic in Computer Science,Mathematics - Functional Analysis